立体几何¶

表面积、体积与直观图¶

定义 1. 常见几何体（人教A必修二P97）¶

多面体：由若干平面多边形围成的几何体叫做多面体. 旋转体：一条平面曲线（包括直线）绕它所在平面内的一条定直线旋转所形成的曲面叫做旋转面，封闭的旋转面围成

多面体：由若干平面多边形围成的几何体叫做多面体. 旋转体：一条平面曲线（包括直线）绕它所在平面内的一条定直线旋转所形成的曲面叫做旋转面，封闭的旋转面围成的几何体叫做旋转体.这条定直线叫做旋转体的轴．棱柱：有两个面互相平行，其余各面都是四边形，并且相邻两个四边形的公共边都互相平行，由这些面所围成的多面体叫做棱柱.两个互相平行的面叫做棱柱的底面，是全等的多边形；其余各面叫做棱柱的侧面，是平行四边形；相邻侧面的公共边叫做侧棱；侧面与底面的公共顶点叫做顶点．侧棱垂直于底面的棱柱叫做直棱柱，侧棱不垂直于底面的棱柱叫做斜棱柱．底面是正多边形的直棱柱叫做正棱柱．底面是平行四边形的四棱柱也叫做平行六面体. 棱锥：有一个面是多边形，其余各面都是有一个公共顶点的三角形，由这些面所围成的多面体叫做棱锥．多边形面叫做底面；有公共顶点的各个三角形面叫做侧面；相邻侧面的公共边叫做侧棱；各侧面的公共顶点叫做顶点．三棱锥又叫四面体．正棱锥是底面是正多边形，并且顶点与底面中心的连线垂直于底面的棱锥．正四面体是侧棱与底面边长相等的正三棱锥，四个面均为等边三角形. 棱台：用一个平行于棱锥底面的平面去截棱锥，把底面和截面之间那部分多面体叫做棱台. 原棱锥的底面和截面分别叫做棱台的下底面和上底面．正棱锥被平行于底面的平面所截，截面与底面之间的部分叫做正棱台，侧面是全等的等腰梯形，侧棱延长后相交于一点. 圆柱：以矩形的一边所在直线为旋转轴，其余三边旋转一周形成的面所围成的旋转体叫做圆柱．旋转轴叫做轴；垂直于轴的边旋转而成的圆面叫做底面；平行于轴的边旋转而成的曲面叫做侧面；无论旋转到什么位置，平行于轴的边都叫做母线．圆柱的上下底及平行于底面的截面均为等圆，轴截面是全等的矩形，侧面展开图为矩形，邻边为底面周长和母线长. 圆锥：以直角三角形的一条直角边所在直线为旋转轴，其余两边旋转一周形成的面所围成的旋转体叫做圆锥. 平行于底面的截面是圆，截面直径与底面直径的比=顶点到截面距离与顶点到底面距离的比. 圆锥轴截面是等腰三角形，母线长满足 l^2 = h^2 + r^2 （ l 为母线， h 为高， r 为底面半径），侧面展开图是以顶点为圆心、母线长为半径的扇形. 圆台：用平行于圆锥底面的平面去截圆锥，底面与截面之间的部分叫做圆台. 球：半圆以它的直径所在直线为旋转轴，旋转一周形成的曲面叫做球面，球面所围成的旋转体叫做球体，简称球．半圆的圆心叫做球心；连接球心和球面上任意一点的线段叫做半径；连接球面上两点并且经过球心的线段叫做直径. 球心与截面圆心的连线垂直于截面，半径满足 R^2 = r^2 + d^2 （ R 为球的半径， r 为截面半径， d 为球心到截面的距离）.

结论 1. 多面体的表面积与体积（人教A必修二P114）¶

结论 2. 旋转体的表面积与体积（人教A必修二P114）¶

定义 2. 斜二测画法（人教A必修二P107）¶

题型 1. 求体积的特殊方法¶

转换顶点：对于三棱锥，任意一个面都可以作为底面.当直接求解某一体积困难，或者需要求解某个点到某平面的距离时，可以通过转换底面和顶点，利用体积相等（ (V_

转换顶点：对于三棱锥，任意一个面都可以作为底面.当直接求解某一体积困难，或者需要求解某个点到某平面的距离时，可以通过转换底面和顶点，利用体积相等（ (V_ C-ABD = V_ A-BCD )）列方程求解. 0.6 x = (-0.3535* cm,-0.3535* cm) , y = ( cm,0cm) , z = (0cm, cm) , % 第一个四面体 V_ C-ABD % 定义点坐标 (B) at (0,0,0); (D) at (0,4,0); (C) at (3,2,0); (A) at (1.5,2,3); % C 到底面 ABD (方程 2x - z = 0) 的高，投影点为 (0.6, 2, 1.2) (Hc) at (0.6,2,1.2); (A) -- (B) -- (C) -- (D) -- cycle; (A) -- (C); [dashed] (B) -- (D); [dashed, thick] (C) -- (Hc); (Hc) circle (1.5pt); [above] at (A) A ; [left] at (B) B ; [right] at (D) D ; [below] at (C) C ; (Mid1) at (0,2,0); at ([yshift=-1.3cm]Mid1) V_ C-ABD ; [font= ] at ([xshift=4cm,yshift=0.5cm]0,0,0) ; % 第二个四面体 V_ A-BCD [xshift=6.5cm] (B) at (0,0,0); (D) at (0,4,0); (C) at (3,2,0); (A) at (1.5,2,3); % A 到底面 BCD (方程 z = 0) 的高，恰好为 A 在 xy 平面的射影 (1.5, 2, 0) (Ha) at (1.5,2,0); (A) -- (B) -- (C) -- (D) -- cycle; (A) -- (C); [dashed] (B) -- (D); [dashed, thick] (A) -- (Ha); (Ha) circle (1.5pt); [above] at (A) A ; [left] at (B) B ; [right] at (D) D ; [below] at (C) C ; (Mid2) at (0,2,0); at ([yshift=-1.3cm]Mid2) V_ A-BCD ; 比例放缩：当两个棱锥具有“共高”或“共底”的性质时，或其底面、高之间存在明确比例关系时，可以直接通过面积比、高之比转化为体积比. [scale=0.8, >=Stealth] % 基础参数 1.5 2.5 0 0 3.5 0 1.2 -1 % T1 [xshift=0cm] (A) at ( , ); (B) at ( , ); (D) at ( , ); (C) at ( , ); (A) -- (B) -- (C) -- (D) -- cycle; (A) -- (C); [dashed] (B) -- (D); [above] at (A) A ; [left] at (B) B ; [right] at (D) D ; [below] at (C) C ; at (1.75,-2) V ; at (4.5, 1) ; % T2 [xshift=5.5cm] (A) at ( , ); (B) at ( , ); (D) at ( , ); (C) at ( , ); (A) -- (B) -- (C) -- (D) -- cycle; (A) -- (C); [dashed] (B) -- (D); (E) at ( (A)!0.5!(B) ); [blue, thick] (E) -- (C); [blue, thick, dashed] (E) -- (D); [left] at (E) E ; [above left] at ( (A)!0.5!(E) ) a ; [left] at ( (E)!0.5!(B) ) a ; [above] at (A) A ; [left] at (B) B ; [right] at (D) D ; [below] at (C) C ; at (1.75,-2) V_ A-ECD = 1 2 V ; at (10, 1) ; % T3 [xshift=11cm] (A) at ( , ); (B) at ( , ); (D) at ( , ); (C) at ( , ); (A) -- (B) -- (C) -- (D) -- cycle; (A) -- (C); [dashed] (B) -- (D); (E) at ( (A)!0.333!(B) ); [blue, thick] (E) -- (C); [blue, thick, dashed] (E) -- (D); [left] at (E) E ; [above left] at ( (A)!0.5!(E) ) a ; [left] at ( (E)!0.5!(B) ) 2a ; [above] at (A) A ; [left] at (B) B ; [right] at (D) D ; [below] at (C) C ; at (1.75,-2) V_ A-ECD = 1 3 V ; at (15.5, 1) ; % T4 [xshift=16.5cm] (A) at ( , ); (B) at ( , ); (D) at ( , ); (C) at ( , ); (A) -- (B) -- (C) -- (D) -- cycle; (A) -- (C); [dashed] (B) -- (D); (E) at ( (A)!0.333!(B) ); (I) at ( (A)!0.667!(C) ); (G) at ( (A)!0.2!(D) ); [blue, thick] (E) -- (I); [blue, thick, dashed] (E) -- (G); [blue, thick, dashed] (I) -- (G); [left] at (E) E ; [below left] at (I) I ; [right] at (G) G ; [above] at (A) A ; [left] at (B) B ; [right] at (D) D ; [below] at (C) C ; [align=center] (details) at (1.75,-3) AE:BE=1:2 ; AI:CI=2:1 AG:DG=1:4 ; V_ A-EIG = 2 45 V ; % 推导公式 [anchor=north, font= ] (formula) at (9.5, -3.8) 推导: V_ A-EIG = 1 3 S_ AEI d_ G-AEI = 1 3 ( 1 3 2 3 S_ ABC ) 1 5 d_ D-ABC = 2 45 1 3 S_ ABC d_ D-ABC = 2 45 V ; 平行线转化与比例转化：利用线段平行得出等高，或者三点共线得出成比例的高，再配合转换顶点快速求体积. [scale=0.8, >=stealth] % 左侧图片 [xshift=1cm] (A) at (0, 3); (Aprime) at (1.5, 3); (B) at (-0.5, 0); (D) at (3, 0); (C) at (0.5, -1.3); % 顶部平行线 [thick] (-1.5,3) -- (3.5,3); (A) -- (B) -- (C) -- (D) -- cycle; (A) -- (C); [dashed] (B) -- (D); % A' 到各顶点的连线 [blue, thick] (Aprime) -- (B); [blue, thick] (Aprime) -- (C); [blue, thick, dashed] (Aprime) -- (D); [above] at (A) A ; [above] at (Aprime) A' ; [left] at (B) B ; [below] at (C) C ; [right] at (D) D ; [align=center] at (1.2, -2.8) 平行转化+转换顶点 AA' BD V_ A-BCD = V_ A'-BCD = V_ C-A'BD ; % 右侧图片 [xshift=9.5cm] % 平面 O B C D (平行四边形框架) (Plane1) at (-3, -1); (Plane2) at (2.5, -1); (Plane3) at (3.8, 1.2); (Plane4) at (-1.7, 1.2); (Plane4) -- (Plane1) -- (Plane2) -- (Plane3); [dashed] (Plane4) -- (Plane3); (O) at (-1.5, 0); (B) at (0, 0); (C) at (1.2, -0.6); (D) at (2.5, 0.4); (P) at (1.2, 3); (A) at ( (O)!0.45!(P) ); % 绘制 O P 线以及相关连线 (O) -- (P); (P) -- (B); (P) -- (C); (P) -- (D); (B) -- (C) -- (D); [dashed] (B) -- (D); % A 的连线 [dashed, blue, thick] (A) -- (B); [blue, thick] (A) -- (C); [dashed, blue, thick] (A) -- (D); [left] at (O) O ; [above] at (P) P ; [left, xshift=-2pt] at (A) A ; [left] at (B) B ; [below] at (C) C ; [right] at (D) D ; [align=center] at (1, -2.8) 比例转化+转换顶点 OA = 1 OP V_ A-BCD = 1 V_ P-BCD = 1 V_ C-PBD ;

外接球问题¶

结论 1. 正方体与长方体¶

0.49 长方体和正方体的外接球的球心为其体对角线的中点，半径为体对角线长的一半．若长方体棱长分别为 a,b,c 则 (R = a^ 2 +b^ 2 +c^ 2 2 ) 若正方体棱长为 a ，则 (R = 3 2 a ) 0.5 % 长方体外接球示意图 1.2 x = (-0.3535* cm,-0.3535* cm) , y = ( cm,0cm) , z = (0cm, cm) , 1.8 % x 2.5 % y 1.8 % z % Define coordinates matching the labels (ABCD base counter-clockwise from front-left?) % Standard Cube: D(back-left), A(front-left), B(front-right), C(back-right) % Our coordinates: % (0,0,0) was O -> D % (a,0,0) was A -> A % (a,b,0) was B -> B % (0,b,0) was C -> C (D) at (0,0,0); (A) at ( ,0,0); (B) at ( , ,0); (C) at (0, ,0); (D1) at (0,0, ); (A1) at ( ,0, ); (B1) at ( , , ); (C1) at (0, , ); % Box - Dashed back lines (from D) [dashed] (D) -- (A); [dashed] (D) -- (C); [dashed] (D) -- (D1); % Box - Visible lines (A) -- (B) -- (C); (D1) -- (A1) -- (B1) -- (C1) -- cycle; (A) -- (A1); (B) -- (B1); (C) -- (C1); % Vertex Labels % [left] at (D) D ; % [left] at (A) A ; % [right] at (B) B ; % [right] at (C) C ; % [left] at (D1) D_1 ; % [left] at (A1) A_1 ; % [right] at (B1) B_1 ; % [right] at (C1) C_1 ; % Diagonal and Center % Was A to C1 (previously G) [dashed, thick] (B) -- (D1) node[midway, above left, black] ; (Center) at ( (B)!0.5!(D1) ); [blue] (Center) circle (1.5pt) node[right, black] O' ; [thick, blue] (Center) -- (B) node[midway, above] R ; % Side Length Labels [right] at ( (B)!0.5!(C) ) b ; [below] at ( (A)!0.5!(B) ) a ; [right] at ( (B)!0.5!(B1) ) c ; 1.2 x = (-0.3535* cm,-0.3535* cm) , y = ( cm,0cm) , z = (0cm, cm) , 1.8 % x 1.8 % y 1.8 % z % Define coordinates matching the labels (ABCD base counter-clockwise from front-left?) % Standard Cube: D(back-left), A(front-left), B(front-right), C(back-right) % Our coordinates: % (0,0,0) was O -> D % (a,0,0) was A -> A % (a,b,0) was B -> B % (0,b,0) was C -> C (D) at (0,0,0); (A) at ( ,0,0); (B) at ( , ,0); (C) at (0, ,0); (D1) at (0,0, ); (A1) at ( ,0, ); (B1) at ( , , ); (C1) at (0, , ); % Box - Dashed back lines (from D) [dashed] (D) -- (A); [dashed] (D) -- (C); [dashed] (D) -- (D1); % Box - Visible lines (A) -- (B) -- (C); (D1) -- (A1) -- (B1) -- (C1) -- cycle; (A) -- (A1); (B) -- (B1); (C) -- (C1); % Vertex Labels % [left] at (D) D ; % [left] at (A) A ; % [right] at (B) B ; % [right] at (C) C ; % [left] at (D1) D_1 ; % [left] at (A1) A_1 ; % [right] at (B1) B_1 ; % [right] at (C1) C_1 ; % Diagonal and Center % Was A to C1 (previously G) [dashed, thick] (B) -- (D1) node[midway, above left, black] ; (Center) at ( (B)!0.5!(D1) ); [blue] (Center) circle (1.5pt) node[right, black] O' ; [thick, blue] (Center) -- (B) node[midway, above] R ; % Side Length Labels % [left] at ( (D)!0.5!(A) ) a ; [below] at ( (A)!0.5!(B) ) a ; % [left] at ( (D)!0.5!(D1) ) c ; 补成长方体或正方体 1 x = (-0.3535* cm,-0.3535* cm) , y = ( cm,0cm) , z = (0cm, cm) , 2 2.5 1.8 (D) at (0,0,0); (A) at ( ,0,0); (B) at ( , ,0); (C) at (0, ,0); (D1) at (0,0, ); (A1) at ( ,0, ); (B1) at ( , , ); (C1) at (0, , ); % Cube (Gray) [gray, dashed] (D) -- (A) -- (B) -- (C) -- cycle; [gray, dashed] (D1) -- (A1) -- (B1) -- (C1) -- cycle; [gray, dashed] (D1) -- (D); [gray, dashed] (A1) -- (A); [gray, dashed] (B1) -- (B); [gray, dashed] (C1) -- (C); % Tetrahedron B-AB1C (Corner B) % BA, BC, BB1 are mutually perpendicular at B [thick, blue] (B) -- (A); [thick, blue] (B) -- (C); [thick, blue] (B) -- (B1); [thick, blue] (A) -- (B1); [thick, blue] (C) -- (B1); [thick, blue, dashed] (A) -- (C); [blue] (B) circle (1.5pt); [blue] (A) circle (1.5pt); [blue] (C) circle (1.5pt); [blue] (B1) circle (1.5pt); 1 x = (-0.3535* cm,-0.3535* cm) , y = ( cm,0cm) , z = (0cm, cm) , 2 2.5 1.8 (D) at (0,0,0); (A) at ( ,0,0); (B) at ( , ,0); (C) at (0, ,0); (D1) at (0,0, ); (A1) at ( ,0, ); (B1) at ( , , ); (C1) at (0, , ); % Cube (Gray) [gray, dashed] (D) -- (A) -- (B) -- (C) -- cycle; [gray, dashed] (D1) -- (A1) -- (B1) -- (C1) -- cycle; [gray, dashed] (D1) -- (D); [gray, dashed] (A1) -- (A); [gray, dashed] (B1) -- (B); [gray, dashed] (C1) -- (C); % Tetrahedron A1-ABC (Similar to simple corner but different faces) % Faces: A1AB(Rt), ABC(Rt), A1AC(Rt), A1BC(Rt) [thick, blue] (A1) -- (A); [thick, blue] (A1) -- (B); [thick, blue] (A) -- (B); [thick, blue] (B) -- (C); [thick, blue, dashed] (A) -- (C); [thick, blue, dashed] (A1) -- (C); [blue] (A1) circle (1.5pt); [blue] (A) circle (1.5pt); [blue] (B) circle (1.5pt); [blue] (C) circle (1.5pt); 1 x = (-0.3535* cm,-0.3535* cm) , y = ( cm,0cm) , z = (0cm, cm) , 2 2.5 1.8 (D) at (0,0,0); (A) at ( ,0,0); (B) at ( , ,0); (C) at (0, ,0); (D1) at (0,0, ); (A1) at ( ,0, ); (B1) at ( , , ); (C1) at (0, , ); % Cube (Gray) [gray, dashed] (D) -- (A) -- (B) -- (C) -- cycle; [gray, dashed] (D1) -- (A1) -- (B1) -- (C1) -- cycle; [gray, dashed] (D1) -- (D); [gray, dashed] (A1) -- (A); [gray, dashed] (B1) -- (B); [gray, dashed] (C1) -- (C); % Tetrahedron D1-ABC (D1 projects to D) [thick, blue] (D1) -- (A); [thick, blue] (D1) -- (B); [thick, blue] (D1) -- (C); % Actually goes "through" back [thick, blue] (A) -- (B); [thick, blue] (B) -- (C); [thick, blue, dashed] (A) -- (C); % D1 is back-top-left. C is back-right. % Check D1-C visibility. Back face diagonal. Usually dashed if back face hidden. % In this view (Front-Top-Right visible), Back face is hidden. [blue] (D1) circle (1.5pt); [blue] (A) circle (1.5pt); [blue] (B) circle (1.5pt); [blue] (C) circle (1.5pt); 1 x = (-0.3535* cm,-0.3535* cm) , y = ( cm,0cm) , z = (0cm, cm) , 2 2.5 1.8 (D) at (0,0,0); (A) at ( ,0,0); (B) at ( , ,0); (C) at (0, ,0); (D1) at (0,0, ); (A1) at ( ,0, ); (B1) at ( , , ); (C1) at (0, , ); % Cube (Gray) [gray, dashed] (D) -- (A) -- (B) -- (C) -- cycle; [gray, dashed] (D1) -- (A1) -- (B1) -- (C1) -- cycle; [gray, dashed] (D1) -- (D); [gray, dashed] (A1) -- (A); [gray, dashed] (B1) -- (B); [gray, dashed] (C1) -- (C); % Tetrahedron A-C-D1-B1 [thick, blue] (A) -- (B1); [above] at ( (A)!0.5!(B1) ) m ; [thick, blue] (C) -- (B1); [right] at ( (C)!0.5!(B1) ) n ; [thick, blue] (D1) -- (B1); [above] at ( (D1)!0.5!(B1) ) t ; [thick, blue, dashed] (A) -- (C); [thick, blue, dashed] (A) -- (D1); [thick, blue, dashed] (C) -- (D1); [blue] (A) circle (1.5pt); [blue] (C) circle (1.5pt); [blue] (D1) circle (1.5pt); [blue] (B1) circle (1.5pt); 若三棱锥的三条侧棱两两互相垂直，则可将其放入某个长方体内．若三棱锥的四个面均是直角三角形，则此时可构造长方体. 若三棱锥的顶点投影在底面构成矩形，则此时可构造长方体. 若三棱锥的对棱两两相等，则可将其放入某个长方体内，对于该四面体, 设长方体的棱长分别为 (a,b,c ) 则 (b^ 2 +c^ 2 =m^ 2 , a^ 2 +c^ 2 =n^ 2 , a^ 2 +b^ 2 =t^ 2 )，三式相加可得 (a^ 2 +b^ 2 +c^ 2 = m^ 2 +n^ 2 +t^ 2 2 )，而显然四面体和长方体有相同的外接球，则 (a^ 2 +b^ 2 +c^ 2 =4R^ 2 )， (R = m^ 2 +n^ 2 +t^ 2 8 ) 特别地，对于正四面体可以补形为正方体，若正四面体的棱长为 a ， R= 6 4 a 对于直角三角形平面，通常可以扩展为矩形.

结论 2. 柱体¶

结论 3. 锥体¶

0.49 对于圆锥模型，侧棱相等的棱锥，可放到圆锥中考虑，有 h - R ^ 2 +r^ 2 =R^ 2 R = h^ 2 +r^ 2 2h ，其中 (h )为高， (r )为底面外接圆半径，此模型要求顶点在底面的投影在底面外接圆的圆心上. 0.5 % Case 1: 球心在圆锥内部 (Center O inside the cone) [scale=1, >=stealth, baseline=(current bounding box.center)] 2 % 外接球半径 -1.2 % 底面圆心 O1 的 y 坐标 (O 在原点, O1 在 O 下方) % 计算圆锥高 h 和底面半径 r - % 顶点 P 在 (0, ), 故高 h = - sqrt( ^2 - ) % 底面半径 r (由勾股定理) % 定义坐标 (O) at (0,0); % 外接球心 (O1) at (0, ); % 圆锥底面圆心 (P) at (0, ); % 圆锥顶点 (A) at ( , ); % 底面圆周上一点 (右侧) (AL) at (- , ); % 底面圆周上一点 (左侧) % 1. 绘制外接球 [ball color=cyan!30, opacity=0.2] (O) circle[radius= ]; (O) circle ( ); % 球的大圆 (正视图) [dashed] (- ,0) arc (180:0: and 0.3 ); % 赤道后弧 (虚线) (- ,0) arc (180:360: and 0.3 ); % 赤道前弧 (实线) % 2. 绘制圆锥底面 (椭圆, 模拟 3D 透视) [dashed] (AL) arc (180:0: and 0.2 ); % 底面后半圆 (虚线) (AL) arc (180:360: and 0.2 ); % 底面前半圆 (实线) % 3. 绘制圆锥母线 (AL) -- (P); (A) -- (P); % 4. 绘制辅助线 (蓝色虚线) [dashed, blue] (O1) -- (O) node[midway, left] h-R ; % 圆锥的高 [dashed, blue] (O1) -- (A) node[midway, below] r ; % 底面半径 [dashed, blue] (O) -- (A) node[midway, above] R ; % 外接球半径 [dashed] (O) -- (P); % 外接球半径 (到顶点) % 5. 标记关键点 (O) circle (1.5pt) node[left] O ; (O1) circle (1.5pt) node[left] O_1 ; % (P) circle (1.5pt) node[above] P ; [right] at (A) A ; [scale=1, >=stealth, baseline=(current bounding box.center)] 2 % 外接球半径 0.8 % 底面圆心 O1 的 y 坐标 (O 在原点, O1 在 O 上方) % 计算圆锥高 h 和底面半径 r - % 顶点 P 在 (0, ), 故高 h = - sqrt( ^2 - ^2) % 底面半径 r (由勾股定理) % 定义坐标 (O) at (0,0); % 外接球心 (O1) at (0, ); % 圆锥底面圆心 (P) at (0, ); % 圆锥顶点 (A) at ( , ); % 底面圆周上一点 (右侧) (AL) at (- , ); % 底面圆周上一点 (左侧) % 1. 绘制外接球 [ball color=cyan!30, opacity=0.2] (O) circle[radius= ]; (O) circle ( ); % 球的大圆 (正视图) [dashed] (- ,0) arc (180:0: and 0.3 ); % 赤道后弧 (虚线) (- ,0) arc (180:360: and 0.3 ); % 赤道前弧 (实线) % 2. 绘制圆锥底面 (椭圆, 模拟 3D 透视) [dashed] (AL) arc (180:0: and 0.2 ); % 底面后半圆 (虚线) (AL) arc (180:360: and 0.2 ); % 底面前半圆 (实线) % 3. 绘制圆锥母线 (AL) -- (P); (A) -- (P); % 4. 绘制辅助线 (蓝色虚线) [dashed] (O1) -- (P) node[midway, left] h ; % 圆锥的高 [dashed, blue] (O1) -- (A) node[midway, above] r ; % 底面半径 [dashed, blue] (O) -- (A) node[midway, below] R ; % 外接球半径 % [dashed, blue] (O) -- (P) node[midway, left] R ; % 外接球半径 (到顶点) [dashed, blue] (O) -- (O1) node[midway, left] R-h ; % 球心到底面的距离 % 5. 标记关键点 (O) circle (1.5pt) node[left] O ; (O1) circle (1.5pt) node[left] O_1 ; % (P) circle (1.5pt) node[above] P ; [right] at (A) A ;

结论 4. 台体¶

0.49 设圆台的上、下底面半径分别为 (r_1 )， (r_2 )，高 (O_1O_2 = h )，左图满足方程 ( R^ 2 -r_ 1 ^ 2 + R^ 2 -r_ 2 ^ 2 =h ) 右图满足方程 ( R^ 2 -r_ 1 ^ 2 - R^ 2 -r_ 2 ^ 2 =h ) 事实上，根据上面两个式子，可以化简得： R^2= [(r_1+r_2)^2+h^2] [(r_1-r_2)^2+h^2 ] 4h^2 在该结论中令 r_2=0 或 r_2=r_1 ，可得锥体与柱体的结论.棱台模型也可转化为圆台模型处理，只需求出棱台的上、下底面外接圆半径，即可放入圆台分析. 0.5 % 台体外接球示意图 - 两个情况 [scale=1, >=stealth, baseline=(current bounding box.center)] % Case 1: 球心在两圆之间 (Center O between planes O1 and O2) % 保证 A, B 在圆上: x^2 + y^2 = R^2 2 50 % Top plane angle relative to center -30 % Bottom plane angle % Calculate y = R sin(theta), r = R cos(theta) sin( ) sin( ) cos( ) cos( ) (O) at (0,0); (O1) at (0, ); (O2) at (0, ); % Circle points (A) at ( , ); (B) at ( , ); (AL) at (- , ); (BL) at (- , ); % Sphere outline [ball color=cyan!30, opacity=0.2] (O) circle[radius= ]; (O) circle ( ); [dashed] (- ,0) arc (180:0: and 0.3 ); (- ,0) arc (180:360: and 0.3 ); % Base 1 (Top) - Visible [dashed] (AL) arc (180:0: and 0.2 ); (AL) arc (180:360: and 0.2 ); % Base 2 (Bottom) - Partially dashed [dashed] (BL) arc (180:0: and 0.3 ); (BL) arc (180:360: and 0.3 ); % Sides (AL) -- (BL); (A) -- (B); % Internal Lines [dashed, blue] (O1) -- (O2) node[midway, left] h ; % Axis [dashed, blue] (O1) -- (A) node[midway] r_1 ; [dashed, blue] (O2) -- (B) node[midway] r_2 ; [dashed, blue] (O) -- (A) node[midway] R ; [dashed, blue] (O) -- (B) node[midway] R ; % Points (O) circle (1.5pt) node[left] O ; (O1) circle (1.5pt) node[left] O_1 ; (O2) circle (1.5pt) node[left] O_2 ; [above, right] at (A) A ; [right] at (B) B ; % [scale=1, >=stealth, baseline=(current bounding box.center)] % Case 2: 球心在同侧 2 1.5 0.6 % 计算半径 sqrt( ^2 - ^2) sqrt( ^2 - ^2) (O) at (0,0); (O1) at (0, ); (O2) at (0, ); (A) at ( , ); (B) at ( , ); % Sphere [ball color=cyan!30, opacity=0.2] (O) circle[radius= ]; (O) circle ( ); [dashed] (- ,0) arc (180:0: and 0.3 ); (- ,0) arc (180:360: and 0.3 ); % Base 1 (Top) (O1) ellipse ( and 0.2 ); % Base 2 (Mid) (O2) ellipse ( and 0.3 ); % Sides (- , ) -- (- , ); ( , ) -- ( , ); % Internal Lines [dashed, blue] (O) -- (O2); % Axis [dashed, blue] (O1) -- (A) node[midway] r_1 ; [dashed, blue] (O2) -- (B) node[midway] r_2 ; [dashed, blue] (O) -- (A) node[midway] R ; [dashed, blue] (O) -- (B) node[midway] R ; [dashed, blue] (O1) -- (O2) node[midway, left] h ; % Axis % Points (O) circle (1.5pt) node[below] O ; (O1) circle (1.5pt) node[left] O_1 ; (O2) circle (1.5pt) node[left] O_2 ; [above, right] at (A) A ; [right] at (B) B ;

结论 5. 二面角模型¶

0.49 [>=stealth, scale=0.85, font= , x= ( -0.1cm , 0.4cm ) , y= ( 1cm , 0cm ) , z= ( 0cm , 1cm ) ] % --- 几何基础参数设置 (随时可调) --- 2 % M到A的距离 (即AB的一半) 2.5 % M到O1的距离 2.3 % M到O2的距离 70 % 二面角的大小 (修改此处即可调整O2和外接圆的张角) % --- 自动推算三维坐标与半径 --- * cos( ) * sin( ) ( - * cos( )) / sin( ) sqrt( + ) sqrt( + ) sqrt( + + ) % 真正的三维几何坐标 (M为原点, AB所在直线为x轴, 面ABD在xy平面, 面ABC沿x轴以二面角旋转) (M) at (0,0,0); (A) at ( ,0,0); (B) at (- ,0,0); (O1) at (0, ,0); (O2) at (0, , ); (O) at (0, , ); % 外接球的光照风格球体与轮廓 [x= (1cm,0cm) , y= (0cm,1cm) ] [ball color=cyan!30, opacity=0.2] (O) circle[radius= cm]; [cyan, thick, densely dashed] (O) circle[radius= cm]; % 改变这里的角度即可轻松调整C和D在圆周上的位置 100 90 (C) at ( cos( ) , + cos( )*sin( ) , + sin( )*sin( ) ); % 面ABC上的点 (D) at ( cos( ) , + sin( ) , 0); % 面ABD上的点 % 椭圆1 (平面ABD) [fill=blue!8, fill opacity=0.7, dashed, draw=blue!70!black] plot[domain=0:360, samples=80, smooth] ( cos( ) , + sin( ) , 0); % 椭圆2 (平面ABC) [fill=blue!8, fill opacity=0.7, dashed, draw=blue!70!black] plot[domain=0:360, samples=80, smooth] ( cos( ) , + cos( )*sin( ) , + sin( )*sin( ) ); % 辅助线 (虚线) [dashed] (M) -- (O1); [dashed] (M) -- (O2); [dashed] (O) -- (O1); [dashed] (O) -- (O2); [dashed] (O) -- (M); % 几何体轮廓 (实线) (A) -- (B); (C) -- (D); (B) -- (C) -- (A); (B) -- (D) -- (A); % 直角标记 O1 (A1) at ( (O1)!0.25cm!(M) ); (B1) at ( (O1)!0.25cm!(O) ); [line join=round] (A1) -- ( (A1)+(B1)-(O1) ) -- (B1); % 直角标记 O2 (A2) at ( (O2)!0.25cm!(M) ); (B2) at ( (O2)!0.25cm!(O) ); [line join=round] (A2) -- ( (A2)+(B2)-(O2) ) -- (B2); % 交点与标签 (M) circle (1.5pt) node[left] M ; (O) circle (1.5pt) node[above right] O ; (O1) circle (1.5pt) node[below right, xshift=-2pt] O_1 ; (O2) circle (1.5pt) node[left, xshift=-2pt] O_2 ; (A) circle (1.5pt) node[above] A ; (B) circle (1.5pt) node[below left] B ; (C) circle (1.5pt) node[left] C ; (D) circle (1.5pt) node[right] D ; 0.49 [>=stealth, scale=0.85, font= , x= ( 0.1cm , 0.4cm ) , y= ( 1cm , 0cm ) , z= ( 0cm , 1cm ) ] % --- 几何基础参数设置 (随时可调) --- 1.5 % M到A的距离 (即AB的一半) 2.0 % M到O1的距离 1.8 % M到O2的距离 110 % 二面角的大小 (修改此处即可调整O2和外接圆的张角) % --- 自动推算三维坐标与半径 --- * cos( ) * sin( ) ( - * cos( )) / sin( ) sqrt( + ) sqrt( + ) sqrt( + + ) % 真正的三维几何坐标 (M为原点, AB所在直线为x轴, 面ABD在xy平面, 面ABC沿x轴以二面角旋转) (M) at (0,0,0); (A) at ( ,0,0); (B) at (- ,0,0); (O1) at (0, ,0); (O2) at (0, , ); (O) at (0, , ); % 外接球的光照风格球体与轮廓 [x= (1cm,0cm) , y= (0cm,1cm) ] [ball color=cyan!30, opacity=0.2] (O) circle[radius= cm]; [cyan, thick, densely dashed] (O) circle[radius= cm]; % 改变这里的角度即可轻松调整C和D在圆周上的位置 100 90 (C) at ( cos( ) , + cos( )*sin( ) , + sin( )*sin( ) ); % 面ABC上的点 (D) at ( cos( ) , + sin( ) , 0); % 面ABD上的点 % 椭圆1 (平面ABD) [fill=blue!8, fill opacity=0.7, dashed, draw=blue!70!black] plot[domain=0:360, samples=80, smooth] ( cos( ) , + sin( ) , 0); % 椭圆2 (平面ABC) [fill=blue!8, fill opacity=0.7, dashed, draw=blue!70!black] plot[domain=0:360, samples=80, smooth] ( cos( ) , + cos( )*sin( ) , + sin( )*sin( ) ); % 辅助线 (虚线) [dashed] (M) -- (O1); [dashed] (M) -- (O2); [dashed] (O) -- (O1); [dashed] (O) -- (O2); [dashed] (O) -- (M); % 几何体轮廓 (实线) (A) -- (B); (C) -- (D); (B) -- (C) -- (A); (B) -- (D) -- (A); % 直角标记 O1 (A1) at ( (O1)!0.25cm!(M) ); (B1) at ( (O1)!0.25cm!(O) ); [line join=round] (A1) -- ( (A1)+(B1)-(O1) ) -- (B1); % 直角标记 O2 (A2) at ( (O2)!0.25cm!(M) ); (B2) at ( (O2)!0.25cm!(O) ); [line join=round] (A2) -- ( (A2)+(B2)-(O2) ) -- (B2); % 交点与标签 (M) circle (1.5pt) node[left] M ; (O) circle (1.5pt) node[above right] O ; (O1) circle (1.5pt) node[below right, xshift=-2pt] O_1 ; (O2) circle (1.5pt) node[left, xshift=-2pt] O_2 ; (A) circle (1.5pt) node[above] A ; (B) circle (1.5pt) node[below left] B ; (C) circle (1.5pt) node[left] C ; (D) circle (1.5pt) node[right] D ; 0.7 给定三棱锥 C-ABD ，已知二面角 (C-AB-D )，则外接球半径 (R )满足： [ R^ 2 = ( O_1O_2 )^2+ ( l 2 )^2= m^ 2 +n^ 2 -2mn ^ 2 + l^ 2 4 ] 0.29 [>=stealth, scale=0.85] 1.8 (C) at (0,0); (M) at (- , 0); (O) at ( , 0); (C) circle ( ); (C) circle (1pt); 112 238 (O2) at ( : ); (O1) at ( : ); (M) -- (O); (M) -- (O2) -- (O); (M) -- (O1) -- (O); (O1) -- (O2); % 直角标记 O2 (A2) at ( (O2)!0.2cm!(M) ); (B2) at ( (O2)!0.2cm!(O) ); (A2) -- ( (A2)+(B2)-(O2) ) -- (B2); % 直角标记 O1 (A1) at ( (O1)!0.2cm!(M) ); (B1) at ( (O1)!0.2cm!(O) ); (A1) -- ( (A1)+(B1)-(O1) ) -- (B1); % 夹角 theta [blue, semithick] ( (M) + (-59:0.4) ) arc (-59:56:0.4); at ( (M) + (20:0.6) ) ; (M) circle (1.2pt) node[left] M ; (O) circle (1.2pt) node[right] O ; (O1) circle (1.2pt) node[below] O_1 ; (O2) circle (1.2pt) node[above left, xshift=2pt] O_2 ; 其中 M 为 AB 中点， (l )为两平面的交线长度 (AB ) ， (m ) 为 ( ABD )外接圆圆心 O_1 到交线距离 (O_ 1 M )， (n ) 为 ( ABC )外接圆圆心 O_2 到交线距离 (O_ 2 M )， ( ) 为 O_1MO_2 即 (m )与 (n )夹角，当且仅当 ACB 和 ADB 有一个为钝角时，为二面角的补角，其他情况即为二面角 (C-AB-D ) 证明：在 O_1MO_2 中由余弦定理得： O_ 1 O_ 2 ^ 2 =O_ 1 M^ 2 +O_ 2 M^ 2 -2O_ 1 M O_ 2 M =m^ 2 +n^ 2 -2mn 易知以 (OM )为直径的圆经过 (O_ 1 ,O_ 2 )，在 O_1MO_2 中由正弦定理得: OM = O_ 1 O_ 2 = m^ 2 +n^ 2 -2mn 在 (Rt OMA )中使用勾股定理求外接球半径 R 即求 OA ， OA^2=R^ 2 = m^ 2 +n^ 2 -2mn ^ 2 + l^ 2 4

内切球问题¶

结论 1. 内切球通用结论¶

结论 2. 圆柱与直棱柱的内切球¶

圆柱：当且仅当圆柱的轴截面为正方形（即高等于底面直径 (h=2r )）时，圆柱有内切球. 此时，内切球的半径 (R ) 既等于圆柱底面半径 (r )，也

结论 3. 圆锥的内切球¶

0.65 圆锥的轴截面为等腰三角形，等腰三角形的内切圆为内切球的大圆，内切圆的半径即为内切球的半径，设圆锥底面半径为 (r )，高为 (h )，则 [ S_

结论 4. 正棱锥的内切球¶

0.65 三棱锥一定有内切球，但四棱锥及以上不一定有内切球. 对于正四、六、八棱锥，通过底面得边中点的轴截面的内切圆为棱锥内切球的大圆，内切圆的半径为内切球的

结论 5. 圆台的内切球存在性¶

0.65 存在内切球的圆台满足如下性质： [ 内切球半径为上下底面圆半径的等比中项 R^2 = r_1 r_2 圆台高对应内切球的直径 h = 2R 圆台母线为

四面体与正方体¶

性质 1. 四面体的性质¶

三组对棱分别相等的四面体必内接于唯一的长方体，且四面体的棱分别为长方体的面对角线；若四面体有三条棱两两互相垂直，则可将其放入某个长方体内；若四面体的四个面均

结论 1. 正四面体的性质结论¶

性质 2. 正方体的性质¶

0.6 正方体的体对角线 ( BD_1 ) 平面 ( A_1C_1D )，且 ( BD_1 ) 平面 ( ACB_1 )； ( F ) 为 ( A_1C_1D

截面问题¶

题型 1. 扩大截面¶

扩大截面常用平行线法和延长线法来寻找多面体的完整截面平行线法：过截面一顶点作对边的平行线延长线法：延长截面中位于几何体表面的线，使之与几何体的棱相交 0.49 [scale=1.1, >=stealth, line join=round, line cap=round, x= (-0.3535cm,-0.3535cm) , y= (1cm,0cm) , z= (0cm,1cm) ] % 左半边：图1（原始状态） (D1) at (0,0,0); (A1) at (2,0,0); (B1) at (2,2,0); (C1) at (0,2,0); (D) at (0,0,2); (A) at (2,0,2); (B) at (2,2,2); (C) at (0,2,2); (E) at (2,2,1); [dashed] (A1) -- (D1) -- (C1); [dashed] (D1) -- (D); (A1) -- (B1) -- (C1); (A) -- (A1); (B) -- (B1); (C) -- (C1); (A) -- (B) -- (C) -- (D) -- cycle; [dashed] (A1) -- (D); [dashed] (D) -- (E); [thick] (A1) -- (E); [below left] at (A1) A ; [below right] at (B1) B ; [right] at (C1) C ; [left] at (D1) D ; [left] at (A) A_1 ; [above left] at (B) B_1 ; [right] at (C) C_1 ; [left] at (D) D_1 ; [right] at (E) E ; % 右半边：图2（补全截面，仅新增线段为蓝色） [xshift=3.5cm] (D1) at (0,0,0); (A1) at (2,0,0); (B1) at (2,2,0); (C1) at (0,2,0); (D) at (0,0,2); (A) at (2,0,2); (B) at (2,2,2); (C) at (0,2,2); (E) at (2,2,1); (F) at (1,2,2); [dashed] (A1) -- (D1) -- (C1); [dashed] (D1) -- (D); (A1) -- (B1) -- (C1); (A) -- (A1); (B) -- (B1); (C) -- (C1); (A) -- (B) -- (C) -- (D) -- cycle; % 图1中已有的线段保持黑色（在内部的依然画虚线，在外部的画实线） [dashed] (A1) -- (D); [dashed] (D) -- (E); [thick] (A1) -- (E); % 新增的补全截面的线段用蓝色表示 [blue, thick] (E) -- (F); [blue, thick] (F) -- (D); [below left] at (A1) A ; [below right] at (B1) B ; [right] at (C1) C ; [left] at (D1) D ; [left] at (A) A_1 ; [above left] at (B) B_1 ; [right] at (C) C_1 ; [left] at (D) D_1 ; [right] at (E) E ; [below] at (F) F ; 0.50 [scale=1.1, >=stealth, line join=round, line cap=round, x= (-0.3535cm,-0.3535cm) , y= (1cm,0cm) , z= (0cm,1cm) ] % 左图：截面初始态与辅助线 (D) at (0,0,0); (A) at (2,0,0); (B) at (2,2,0); (C) at (0,2,0); (D1) at (0,0,2); (A1) at (2,0,2); (B1) at (2,2,2); (C1) at (0,2,2); (M) at (0,1,2); (N) at (1,2,2); [dashed] (A) -- (D) -- (C); [dashed] (D) -- (D1); (A) -- (B) -- (C); (A) -- (A1); (B) -- (B1); (C) -- (C1); (A1) -- (B1) -- (C1) -- (D1) -- cycle; [dashed] (A) -- (M); [dashed] (A) -- (N); (M) -- (N); [blue, thick] ( (A) - (1,1,0) ) -- ( (A) + (0.8,0.8,0) ) node[right] l ; [below left] at (A) A ; [below right] at (B) B ; [right] at (C) C ; [left] at (D) D ; [left] at (A1) A_1 ; [above left] at (B1) B_1 ; [right] at (C1) C_1 ; [left] at (D1) D_1 ; [above] at (M) M ; [right] at (N) N ; % 右图：延长线法完全成型的截面多边形 [xshift=3.5cm] (D) at (0,0,0); (A) at (2,0,0); (B) at (2,2,0); (C) at (0,2,0); (D1) at (0,0,2); (A1) at (2,0,2); (B1) at (2,2,2); (C1) at (0,2,2); (M) at (0,1,2); (N) at (1,2,2); (H) at (2,3,2); (P) at (2,2,1.333); (G) at (0,0,1.333); [dashed] (A) -- (D) -- (C); [dashed] (D) -- (D1); (A) -- (B) -- (C); (A) -- (A1); (B) -- (B1); (C) -- (C1); (A1) -- (B1) -- (C1) -- (D1) -- cycle; % 图1中已有的线段保持黑色 (M) -- (N); [dashed] (A) -- (M); [dashed] (A) -- (N); % 补全截面的延长线及新增轮廓线用蓝色表示 [blue, thick] (B1) -- (H); % 延长立体几何主边作辅助线 [blue, thick] (N) -- (P); [blue, thick] (P) -- (A); [blue, dashed, thick] (A) -- (G); [blue, dashed, thick] (G) -- (M); [blue, thick] (N) -- (H) -- (P); % 延伸到外部的三角形辅助部分 [below left] at (A) A ; [below right] at (B) B ; [right] at (C) C ; [left] at (D) D ; [left] at (A1) A_1 ; [above left] at (B1) B_1 ; [right] at (C1) C_1 ; [left] at (D1) D_1 ; [above] at (M) M ; [above] at (N) N ; [right] at (H) H ; [right] at (P) P ; [left] at (G) G ;

结论 1. \textbf{正方体的截面}¶

只能是三角形、四边形、五边形、六边形. 当截面是三角形时，不可能是直角三角形和钝角三角形；当截面是四边形时，则四边形至少有一组对边平行，不可能是直角梯形；当

只能是三角形、四边形、五边形、六边形. 当截面是三角形时，不可能是直角三角形和钝角三角形；当截面是四边形时，则四边形至少有一组对边平行，不可能是直角梯形；当截面是五边形时，则必有两组分别平行的边，不可能是正五边形；当截面是六边形时，则必有三组平行的边，特别地，可以是正六边形. 正方体截面面积最大时，截面为体对角面（矩形） % Row 1 0.24 [scale=1, >=stealth, line join=round, line cap=round] x= (-0.3535cm,-0.3535cm) , y= (1cm,0cm) , z= (0cm,1cm) 2 (D) at (0,0,0); (A) at ( ,0,0); (B) at ( , ,0); (C) at (0, ,0); (D1) at (0,0, ); (A1) at ( ,0, ); (B1) at ( , , ); (C1) at (0, , ); % Cut Points: Triangle near C1 (P1) at ( , 1.3, ); % on A1B1 (approx) (P2) at (0.7, , ); % on C1B1 (approx) (P3) at ( , , 1.3); % on B1B (approx) % Fill Section [gray!40] (P1) -- (P2) -- (P3) -- cycle; (P1) -- (P2) -- (P3) -- cycle; % Cube skeleton [dashed] (D) -- (A); [dashed] (D) -- (C); [dashed] (D) -- (D1); (A) -- (B) -- (C); (A) -- (A1); (B) -- (B1); (C) -- (C1); (D1) -- (A1) -- (B1) -- (C1) -- cycle; % Labels [left] at (D) D ; [left] at (A) A ; [right] at (B) B ; [right] at (C) C ; [left] at (D1) D_1 ; [left] at (A1) A_1 ; [right] at (B1) B_1 ; [right] at (C1) C_1 ; % 0.24 [scale=1, >=stealth, line join=round, line cap=round] x= (-0.3535cm,-0.3535cm) , y= (1cm,0cm) , z= (0cm,1cm) 2 (D) at (0,0,0); (A) at ( ,0,0); (B) at ( , ,0); (C) at (0, ,0); (D1) at (0,0, ); (A1) at ( ,0, ); (B1) at ( , , ); (C1) at (0, , ); % Cut Points: Triangle A1-B-C1 [gray!40] (A1) -- (B) -- (C1) -- cycle; (A1) -- (B) -- (C1) -- cycle; [dashed] (D) -- (A); [dashed] (D) -- (C); [dashed] (D) -- (D1); (A) -- (B) -- (C); (A) -- (A1); (B) -- (B1); (C) -- (C1); (D1) -- (A1) -- (B1) -- (C1) -- cycle; [left] at (D) D ; [left] at (A) A ; [right] at (B) B ; [right] at (C) C ; [left] at (D1) D_1 ; [left] at (A1) A_1 ; [right] at (B1) B_1 ; [right] at (C1) C_1 ; % 0.24 [scale=1, >=stealth, line join=round, line cap=round] x= (-0.3535cm,-0.3535cm) , y= (1cm,0cm) , z= (0cm,1cm) 2 (D) at (0,0,0); (A) at ( ,0,0); (B) at ( , ,0); (C) at (0, ,0); (D1) at (0,0, ); (A1) at ( ,0, ); (B1) at ( , , ); (C1) at (0, , ); % Cut Points: Triangle near B1 (P1) at ( , 1, ); % on B1A1? No, A1B1 is y changes. (2, y, 2). y=1.3. (P2) at (1, , ); % on B1C1? (x, 2, 2). x=1.3. (P3) at ( , , 1); % on B1B. [gray!40] (P1) -- (P2) -- (P3) -- cycle; (P1) -- (P2) -- (P3) -- cycle; [dashed] (D) -- (A); [dashed] (D) -- (C); [dashed] (D) -- (D1); (A) -- (B) -- (C); (A) -- (A1); (B) -- (B1); (C) -- (C1); (D1) -- (A1) -- (B1) -- (C1) -- cycle; [left] at (D) D ; [left] at (A) A ; [right] at (B) B ; [right] at (C) C ; [left] at (D1) D_1 ; [left] at (A1) A_1 ; [right] at (B1) B_1 ; [right] at (C1) C_1 ; % 0.24 [scale=1, >=stealth, line join=round, line cap=round] x= (-0.3535cm,-0.3535cm) , y= (1cm,0cm) , z= (0cm,1cm) 2 (D) at (0,0,0); (A) at ( ,0,0); (B) at ( , ,0); (C) at (0, ,0); (D1) at (0,0, ); (A1) at ( ,0, ); (B1) at ( , , ); (C1) at (0, , ); % Cut Points: Rhombus A-M-C1-N (M) at ( , 1, 0); % mid AB (N) at (0, 1, ); % mid CD1 [gray!40] (A1) -- (M) -- (C) -- (N) -- cycle; (A1) -- (M) -- (C) -- (N) -- cycle; [dashed] (D) -- (A); [dashed] (D) -- (C); [dashed] (D) -- (D1); (A) -- (B) -- (C); (A) -- (A1); (B) -- (B1); (C) -- (C1); (D1) -- (A1) -- (B1) -- (C1) -- cycle; [left] at (D) D ; [left] at (A) A ; [right] at (B) B ; [right] at (C) C ; [left] at (D1) D_1 ; [left] at (A1) A_1 ; [right] at (B1) B_1 ; [right] at (C1) C_1 ; % 0.24 [scale=1, >=stealth, line join=round, line cap=round] x= (-0.3535cm,-0.3535cm) , y= (1cm,0cm) , z= (0cm,1cm) 2 (D) at (0,0,0); (A) at ( ,0,0); (B) at ( , ,0); (C) at (0, ,0); (D1) at (0,0, ); (A1) at ( ,0, ); (B1) at ( , , ); (C1) at (0, , ); (P1) at ( , 0, 1); % on A1A (P2) at ( , 1, 0); % on AB % Cut Points: Rectangle A1-B-C-D1 [gray!40] (P1) -- (P2) -- (C) -- (D1) -- cycle; (P1) -- (P2) -- (C) -- (D1) -- cycle; [dashed] (D) -- (A); [dashed] (D) -- (C); [dashed] (D) -- (D1); (A) -- (B) -- (C); (A) -- (A1); (B) -- (B1); (C) -- (C1); (D1) -- (A1) -- (B1) -- (C1) -- cycle; [left] at (D) D ; [left] at (A) A ; [right] at (B) B ; [right] at (C) C ; [left] at (D1) D_1 ; [left] at (A1) A_1 ; [right] at (B1) B_1 ; [right] at (C1) C_1 ; % 0.24 [scale=1, >=stealth, line join=round, line cap=round] x= (-0.3535cm,-0.3535cm) , y= (1cm,0cm) , z= (0cm,1cm) 2 (D) at (0,0,0); (A) at ( ,0,0); (B) at ( , ,0); (C) at (0, ,0); (D1) at (0,0, ); (A1) at ( ,0, ); (B1) at ( , , ); (C1) at (0, , ); % Cut Points: Rectangle A-B-C1-D1 (Plane x+z=2) [gray!40] (A1) -- (A) -- (C) -- (C1) -- cycle; (A1) -- (A) -- (C) -- (C1) -- cycle; [dashed] (D) -- (A); [dashed] (D) -- (C); [dashed] (D) -- (D1); (A) -- (B) -- (C); (A) -- (A1); (B) -- (B1); (C) -- (C1); (D1) -- (A1) -- (B1) -- (C1) -- cycle; [left] at (D) D ; [left] at (A) A ; [right] at (B) B ; [right] at (C) C ; [left] at (D1) D_1 ; [left] at (A1) A_1 ; [right] at (B1) B_1 ; [right] at (C1) C_1 ; 0.24 [scale=1, >=stealth, line join=round, line cap=round] x= (-0.3535cm,-0.3535cm) , y= (1cm,0cm) , z= (0cm,1cm) 2 (D) at (0,0,0); (A) at ( ,0,0); (B) at ( , ,0); (C) at (0, ,0); (D1) at (0,0, ); (A1) at ( ,0, ); (B1) at ( , , ); (C1) at (0, , ); % Pentagon: % Pt1(0.5, 2, 0) on BC. Pt2(1.5, 0, 0) on AD. Pt3(0.5, 0, 2) on D1A1. Pt4(0, 1, 2) on C1D1. Pt5(0, 2, 1) on CC1. (P1) at (0.5, , 0); (P2) at (1.5, 0, 0); (P3) at (0.5, 0, ); (P4) at (0, 1, ); (P5) at (0, , 1); [gray!40] (P1) -- (P5) -- (P4) -- (P3) -- (P2) -- cycle; (P1) -- (P5) -- (P4) -- (P3) -- (P2) -- cycle; [dashed] (D) -- (A); [dashed] (D) -- (C); [dashed] (D) -- (D1); (A) -- (B) -- (C); (A) -- (A1); (B) -- (B1); (C) -- (C1); (D1) -- (A1) -- (B1) -- (C1) -- cycle; [left] at (D) D ; [left] at (A) A ; [right] at (B) B ; [right] at (C) C ; [left] at (D1) D_1 ; [left] at (A1) A_1 ; [right] at (B1) B_1 ; [right] at (C1) C_1 ; % 0.24 [scale=1, >=stealth, line join=round, line cap=round] x= (-0.3535cm,-0.3535cm) , y= (1cm,0cm) , z= (0cm,1cm) 2 (D) at (0,0,0); (A) at ( ,0,0); (B) at ( , ,0); (C) at (0, ,0); (D1) at (0,0, ); (A1) at ( ,0, ); (B1) at ( , , ); (C1) at (0, , ); % Cut Points: Hexagon (Midpoints of edges around AC1 normal) 6 points % Mid AB(2,1,0), BC(1,2,0), CC1(0,2,1), C1D1(0,1,2), D1A1(1,0,2), A1A(2,0,1) (P1) at ( , 1, 0); (P2) at (1, , 0); (P3) at (0, , 1); (P4) at (0, 1, ); (P5) at (1, 0, ); (P6) at ( , 0, 1); [gray!40] (P1) -- (P2) -- (P3) -- (P4) -- (P5) -- (P6) -- cycle; (P1) -- (P2) -- (P3); (P3) -- (P4) -- (P5); (P5) -- (P6) -- (P1); [dashed] (D) -- (A); [dashed] (D) -- (C); [dashed] (D) -- (D1); (A) -- (B) -- (C); (A) -- (A1); (B) -- (B1); (C) -- (C1); (D1) -- (A1) -- (B1) -- (C1) -- cycle; [left] at (D) D ; [left] at (A) A ; [right] at (B) B ; [right] at (C) C ; [left] at (D1) D_1 ; [left] at (A1) A_1 ; [right] at (B1) B_1 ; [right] at (C1) C_1 ;

结论 2. 球面与正方体表面的交线¶

考虑一个球心位于正方体一个顶点的球体与正方体表面的交线，设正方体棱长为 a ，球半径为 r 0.27 若 0 < r a ，球面只与三个平面相交，每个面上

考虑一个球心位于正方体一个顶点的球体与正方体表面的交线，设正方体棱长为 a ，球半径为 r 0.27 若 0 < r a ，球面只与三个平面相交，每个面上的交线为四分之一圆，半径为 r ；若 a < r 2 a ，球面与六个面都相交，与较近平面的交线为半径为 r 的圆弧，与较远平面的交线为半径为 r^2 - a^2 的圆弧； 0.7 1.2 % 初始缩放系数 x = (-0.3535* cm,-0.3535* cm) , % x轴投影方向 y = ( cm,0cm) , % y轴投影方向 z = (0cm, cm) , % z轴投影方向 rotate around z=0 % 绕 z 轴旋转 % 正方体参数 2 % 正方体棱长 1.5 % 球半径 % 正方体顶点坐标（以C为球心） (C) at (0,0,0); % 球心 (D) at ( ,0,0); (A) at ( , ,0); (B) at (0, ,0); (C1) at (0,0, ); (D1) at ( ,0, ); (A1) at ( , , ); (B1) at (0, , ); % 绘制正方体轮廓（底面、顶面、侧棱） % 先画虚线部分（被遮挡的） [dashed] (C) -- (C1); [dashed] (C) -- (D); [dashed] (C) -- (B); % 绘制实线部分 (D) -- (D1) -- (C1) -- (B1) -- (B) -- (A) -- cycle; (A) -- (A1); (D1) -- (A1); (B1) -- (A1); % 标注定点坐标 % [below left] at (C) C ; % [below left] at (D) D ; % [below right] at (A) A ; % [below right] at (B) B ; % [above left] at (C1) C_1 ; % [above left] at (D1) D_1 ; % [above right] at (A1) A_1 ; % [above right] at (B1) B_1 ; % 绘制球心 (C) circle (1.5pt); % 绘制球面与正方体表面的交线 [thick, blue] % 与平面x=0（左侧面）的交线：y² + z² = r² min(90,acos(1/ )*180/pi) [domain=0: ,samples=100,smooth] plot (0, cos( ) , sin( ) ); % 与平面y=0（后侧面）的交线：x² + z² = r² [domain=0: ,samples=100,smooth] plot ( cos( ) , 0, sin( ) ); % 与平面z=0（底面）的交线：x² + y² = r² [domain=0: ,samples=100,smooth] plot ( cos( ) , sin( ) , 0); [shift= (0,3,0) ] 2.4 % 球半径 % 正方体顶点坐标（以C为球心） (C) at (0,0,0); % 球心 (D) at ( ,0,0); (A) at ( , ,0); (B) at (0, ,0); (C1) at (0,0, ); (D1) at ( ,0, ); (A1) at ( , , ); (B1) at (0, , ); % 绘制正方体轮廓（底面、顶面、侧棱） % 先画虚线部分（被遮挡的） [dashed] (C) -- (C1); [dashed] (C) -- (D); [dashed] (C) -- (B); % 绘制实线部分 (D) -- (D1) -- (C1) -- (B1) -- (B) -- (A) -- cycle; (A) -- (A1); (D1) -- (A1); (B1) -- (A1); % 绘制球心 (C) circle (1.5pt); % 计算交线在正方体表面的部分 % 大半径（与x=0,y=0,z=0面相交） sqrt( - ) % 小半径（与x=a,y=a,z=a面相交） % 绘制球面与正方体表面的交线（只在正方体表面上） [thick,gray] % 与平面x=0（左侧面）的交线：y² + z² = r²，但只在正方体范围内 plot[domain=0:90,samples=100,smooth] (0, cos( ) , sin( ) ); % 与平面y=0（后侧面）的交线：x² + z² = r² plot[domain=0:90,samples=100,smooth] ( cos( ) , 0, sin( ) ); % 与平面z=0（底面）的交线：x² + y² = r² plot[domain=0:90,samples=100,smooth] ( cos( ) , sin( ) , 0); % 与平面x=2（右侧面）的交线：y² + z² = r²-4 plot[domain=0:90,samples=100,smooth,red] ( , cos( ) , sin( ) ); % 与平面y=2（前侧面）的交线：x² + z² = r²-4 plot[domain=0:90,samples=100,smooth,red] ( cos( ) , , sin( ) ); % 与平面z=2（顶面）的交线：x² + y² = r²-4 plot[domain=0:90,samples=100,smooth,red] ( cos( ) , sin( ) , ); [thick, blue] % 与平面x=0（左侧面）的交线：y² + z² = r² % 只绘制在正方体表面上的部分：y从0到a，z从0到a atan( / ) % 从与z=a平面的交点开始 90-atan( / ) % 到与y=a平面的交点结束 [domain= : ,samples=100,smooth] plot (0, cos( ) , sin( ) ); % 与平面y=0（后侧面）的交线：x² + z² = r² [domain= : ,samples=100,smooth] plot ( cos( ) , 0, sin( ) ); % 与平面z=0（底面）的交线：x² + y² = r² [domain= : ,samples=100,smooth] plot ( cos( ) , sin( ) , 0); [thick, blue] % 与平面x=a（右侧面）的交线：y² + z² = r²-a² % 计算交点：与z=a平面相交时，y=0；与y=a平面相交时，z=0 [domain=0:90,samples=100,smooth] plot ( , cos( ) , sin( ) ); % 与平面y=a（前侧面）的交线：x² + z² = r²-a² [domain=0:90,samples=100,smooth] plot ( cos( ) , , sin( ) ); % 与平面z=a（顶面）的交线：x² + y² = r²-a² [domain=0:90,samples=100,smooth] plot ( cos( ) , sin( ) , ); [shift= (0,6.5,0) ] 3 % 球半径 % 正方体顶点坐标（以C为球心） (C) at (0,0,0); % 球心 (D) at ( ,0,0); (A) at ( , ,0); (B) at (0, ,0); (C1) at (0,0, ); (D1) at ( ,0, ); (A1) at ( , , ); (B1) at (0, , ); % 绘制正方体轮廓（底面、顶面、侧棱） % 先画虚线部分（被遮挡的） % 先画虚线部分（被遮挡的） [dashed] (C) -- (C1); [dashed] (C) -- (D); [dashed] (C) -- (B); % 绘制实线部分 (D) -- (D1) -- (C1) -- (B1) -- (B) -- (A) -- cycle; (A) -- (A1); (D1) -- (A1); (B1) -- (A1); % 标注定点坐标 % [below left] at (C) C ; % [below left] at (D) D ; % [below right] at (A) A ; % [below right] at (B) B ; % [above left] at (C1) C_1 ; % [above left] at (D1) D_1 ; % [above right] at (A1) A_1 ; % [above right] at (B1) B_1 ; % 绘制球心 (C) circle (1.5pt); % 计算交线在正方体表面的部分 % 大半径（与x=0,y=0,z=0面相交） sqrt( - ) % 小半径（与x=a,y=a,z=a面相交） % 对于蓝色曲线（与x=0,y=0,z=0平面的交线） % 我们需要找到曲线与正方体棱的交点 % 这些交点满足：在x=0平面上，y²+z²=r²，且y=a或z=a acos( / ) % 当z=a时，cosθ = a/r asin( / ) % 当y=a时，sinθ = a/r % 对于红色曲线（与x=a,y=a,z=a平面的交线） % 这些交点满足：在x=a平面上，y²+z²=r²-a²，且y=a或z=a acos( / ) % 当z=a时，cosθ = a/rsmall asin( / ) % 当y=a时，sinθ = a/rsmall % 绘制球面与正方体表面的交线（只在正方体表面上） [thick,gray] % 与平面x=0（左侧面）的交线：y² + z² = r²，但只在正方体范围内 % y和z的范围都是0到a plot[domain=0:90,samples=100,smooth] (0, cos( ) , sin( ) ); % 与平面y=0（后侧面）的交线：x² + z² = r² plot[domain=0:90,samples=100,smooth] ( cos( ) , 0, sin( ) ); % 与平面z=0（底面）的交线：x² + y² = r² plot[domain=0:90,samples=100,smooth] ( cos( ) , sin( ) , 0); % 与平面x=2（右侧面）的交线：y² + z² = r²-4 % 注意：这里需要确保y,z在0到2范围内 plot[domain=0:90,samples=100,smooth,red] ( , cos( ) , sin( ) ); % 与平面y=2（前侧面）的交线：x² + z² = r²-4 plot[domain=0:90,samples=100,smooth,red] ( cos( ) , , sin( ) ); % 与平面z=2（顶面）的交线：x² + y² = r²-4 plot[domain=0:90,samples=100,smooth,red] ( cos( ) , sin( ) , ); [thick, blue] % 与平面x=a（右侧面）的交线：y² + z² = r²-a² [domain= : ,samples=100,smooth] plot ( , cos( ) , sin( ) ); % 与平面y=a（前侧面）的交线：x² + z² = r²-a² [domain= : ,samples=100,smooth] plot ( cos( ) , , sin( ) ); % 与平面z=a（顶面）的交线：x² + y² = r²-a² [domain= : ,samples=100,smooth] plot ( cos( ) , sin( ) , ); 若 2 a < r 3 a ，所有交线都小于90°圆弧，每条弧都从一条棱延伸到相邻的另一条棱；若 r > 3 a ，球面完全包围正方体，无交线.

结论 3. \textbf{圆锥的截面积}¶

圆锥过顶点的截面是一个等腰三角形，当这个截面同时过圆锥的轴时，截面就成了轴截面.在所有过圆锥顶点的截面中，面积最大的不一定是轴截面， 0.5 设圆锥的母线是 (

空间点、直线、平面之间的位置关系¶

定理 1. 基本事实与推论（人教A必修二P124）¶

0.38 过不在一条直线上的三个点，有且只有一个平面. 如果一条直线上的两个点在一个平面内，那么这条直线在这个平面内. 如果两个不重合的平面有一个公共点，那么它

定理 2. 空间中点、直线、平面的位置关系（人教A必修二P124）¶

把不同在任何一个平面内的两条直线叫做异面直线．于是，空间两条直线的位置关系有三种： [ 共面直线：相交直线：在同一平面内，有且只有一个公共点；平行直线：在同

定理 3. 平行关系的判定和性质定理（人教A必修二P133）¶

题型 1. 平行关系的证明¶

平行关系的证明题中，最常见的是证线面平行，以下是三大作辅助线的思路：找平行四边形：可先在面内作一条与已知直线平行的直线，观察构成的图形像不像平行四边形，若像，

平行关系的证明题中，最常见的是证线面平行，以下是三大作辅助线的思路：找平行四边形：可先在面内作一条与已知直线平行的直线，观察构成的图形像不像平行四边形，若像，就尝试去找理由，进行论证即可. 两个重要图形的运用（其原理是上面的线面平行的性质定理，运用时选(1)还是(2)，一般看图就知道） 0.5 点线位于面的两侧：如图5，要证 (AB )，可在 ( )的另一侧尝试找一点 (P )，连接 (PA )， (PB )，则面 (PAB )与 ( )的交线就是我们证线面平行要找的 ( )内的直线. 点线位于面的同侧：如图6，要证 (AB )，可在 ( )的同侧尝试找一点 (P )，连接 (PA )， (PB )，则面 (PAB )与 ( )的交线就是我们证线面平行要找的 ( )内的直线. 0.49 0.9 ! [>=stealth, line join=round, line cap=round] % 图5 [xshift=0cm] (0,0) -- (3,0) -- (3.8, 0.8); (0.8, 0.8) -- (0,0); at (0.4, 0.2) ; % ================= 可调节点 ================= (A) at (0.9, 1.4); (B) at (3.4, 1.4); (P) at (2.0, -0.6); 0.5 % 调节 A' B' 在 PA, PB 上的比例 (0 为 P点, 1 为 A/B点) % ============================================ (A') at ( (P)! !(A) ); (B') at ( (P)! !(B) ); % 自动计算面后方边缘被三角形遮挡的虚线段交点 (I1) at (intersection of 0.8,0.8--3.8,0.8 and P--A); (I2) at (intersection of 0.8,0.8--3.8,0.8 and P--B); (0.8, 0.8) -- (I1); [dashed] (I1) -- (I2); (I2) -- (3.8, 0.8); (A) node[left] A -- (P) node[below] P ; (B) node[right] B -- (P); [thick] (A) -- (B); [thick, blue] (A') -- (B'); at (1.9, -1.5) 图5 ; % 图6 [xshift=4.5cm, yshift=-1cm] (0,0) -- (3,0) -- (3.8, 0.8); (0.8, 0.8) -- (0,0); at (0.4, 0.2) ; % ================= 可调节点 ================= (P) at (2.0, 2.2); (A) at (1.4, 1.2); (B) at (2.6, 1.2); 1.8 % 调节 A' B' 在 PA, PB 延长线上的比例 (大于1则画在AB下方) % ============================================ (A') at ( (P)! !(A) ); (B') at ( (P)! !(B) ); % 自动计算面后方边缘被三角形遮挡的虚线段交点 (I1) at (intersection of 0.8,0.8--3.8,0.8 and P--A'); (I2) at (intersection of 0.8,0.8--3.8,0.8 and P--B'); (0.8, 0.8) -- (I1); [dashed] (I1) -- (I2); (I2) -- (3.8, 0.8); (P) node[above] P -- (A'); (P) -- (B'); [thick] (A) node[left] A -- (B) node[right] B ; [thick, blue] (A') -- (B'); at (1.9, -0.5) 图6 ; % 造面面平行：若前面的两个方法都不易解决问题，那么还可以考虑通过证面面平行，来证明线面平行.

定理 4. 垂直关系的判定和性质（人教A必修二P146）¶

题型 2. 空间中证明垂直关系的常见思路¶

证线面垂直：证明直线垂直于平面内的两条相交直线即可. 证线线垂直：若线与线是共面的，则考虑用平面几何的方法来证；若异面，如图5，要证异面直线 (l )和 (a

结论 1. 常见垂直几何模型¶

every node/.style= font= [scale=0.95] % (1) 等腰梯形 [shift= (0,0) , scale=1.5] (A1)

every node/.style= font= [scale=0.95] % (1) 等腰梯形 [shift= (0,0) , scale=1.5] (A1) at (-1,0); (B1) at (1,0); (D1) at (-0.5,0.866); (C1) at (0.5,0.866); (A1) -- (B1) node[midway, below] 2 -- (C1) node[midway, right] 1 -- (D1) node[midway, above] 1 -- cycle node[midway, left] 1 ; (B1) -- (D1); [below left] at (A1) A ; [below right] at (B1) B ; [above right] at (C1) C ; [above left] at (D1) D ; (P1) at ( (D1)!0.15cm!(A1) ); (Q1) at ( (D1)!0.15cm!(B1) ); (P1) -- ( (P1)+(Q1)-(D1) ) -- (Q1); ( (A1)+(0.3,0) ) arc (0:60:0.3); at ( (A1)+(0.5,0.25) ) 60^ ; at (0,-0.5) AD BD ; % (2) 120^ 菱形 [shift= (5,0) , scale=1.5] (A2) at (0,1); (B2) at (-0.866,0.5); (C2) at (0,0); (D2) at (0.866,0.5); (E2) at ( (C2)!0.5!(D2) ); (A2) -- node[midway, above left] 1 (B2) -- node[midway, below left] 1 (C2) -- node[midway, below right] (D2) -- node[midway, above right] 1 cycle; (A2) -- (E2); [above] at (A2) A ; [left] at (B2) B ; [below] at (C2) C ; [right] at (D2) D ; [below right, inner sep=1pt] at (E2) E ; (P2) at ( (E2)!0.15cm!(A2) ); (Q2) at ( (E2)!0.15cm!(D2) ); (P2) -- ( (P2)+(Q2)-(E2) ) -- (Q2); ( (A2)+(-150:0.25) ) arc (-150:-30:0.25); at ( (A2)+(0,-0.40) ) 120^ ; % ( (C2)!0.50!(E2)+(-0.10,0.10) ) -- ( (C2)!0.50!(E2)+(0.10,-0.10) ); % ( (E2)!0.50!(D2)+(-0.10,0.10) ) -- ( (E2)!0.50!(D2)+(0.10,-0.10) ); at (0,-0.5) AE CD ; % (3) 120^ 平行四边形 [shift= (10,0) , scale=1.5] (A3) at (-1,0); (B3) at ( (A3)+(60:1) ); (D3) at (1,0); (C3) at ( (B3)+(2,0) ); (A3) -- node[midway, left] 1 (B3) -- node[midway, above] 2 (C3) -- node[midway, right] 1 (D3) -- node[midway, below] 2 cycle; (B3) -- (D3); [below left] at (A3) A ; [above left] at (B3) B ; [below right] at (D3) D ; [above right] at (C3) C ; (P3) at ( (B3)!0.15cm!(A3) ); (Q3) at ( (B3)!0.15cm!(D3) ); (P3) -- ( (P3)+(Q3)-(B3) ) -- (Q3); ( (A3)+(0.3,0) ) arc (0:60:0.3); at ( (A3)+(0.5,0.25) ) 60^ ; at (0,-0.5) AB BD ; % (4) 正方形 (或1:1矩形) [shift= (0,-3.5) , scale=1.5] (A4) at (-0.5,0); (B4) at (0.5,0); (C4) at (0.5,1); (D4) at (-0.5,1); (E4) at (-0.5, 0.5); % E 是 AD 中点 (F4) at (0, 1); % F 是 CD 中点 (A4) -- node[midway, below] 1 (B4) -- node[midway, right] 1 (C4) -- node[midway, above] (D4) -- node[midway, left] cycle; (A4) -- (F4); (B4) -- (E4); [below left] at (A4) A ; [below right] at (B4) B ; [above right] at (C4) C ; [above left] at (D4) D ; [left] at (E4) E ; [above] at (F4) F ; (O4) at (intersection of A4--F4 and B4--E4); (P4) at ( (O4)!0.15cm!(A4) ); (Q4) at ( (O4)!0.15cm!(B4) ); (P4) -- ( (P4)+(Q4)-(O4) ) -- (Q4); at (0,-0.5) AF BE ; % (5) 2:3 矩形 [shift= (5,-3.5) , scale=1.5] (A5) at (-0.75, 0); (B5) at (0.75, 0); (C5) at (0.75, 1); (D5) at (-0.75, 1); (E5) at (-0.75, 0.6); (F5) at (-0.35, 1); (H5) at (1.15, 1); (A5) -- node[midway, below] 3 (B5) -- node[midway, right] 2 (C5) -- (D5) -- node[midway, left, shift= (-0.3,0.0) ] 2 cycle; (C5) -- (H5); (A5) -- (F5); (B5) -- (E5); [dashed] (B5) -- (H5) -- (E5); [below left] at (A5) A ; [below right] at (B5) B ; [above] at (C5) C ; [above left] at (D5) D ; [left] at (E5) E ; [above] at (F5) F ; [above right] at (H5) H ; (O5) at (intersection of A5--F5 and B5--E5); (P5) at ( (O5)!0.15cm!(A5) ); (Q5) at ( (O5)!0.15cm!(B5) ); (P5) -- ( (P5)+(Q5)-(O5) ) -- (Q5); (P5_2) at ( (B5)!0.15cm!(H5) ); (Q5_2) at ( (B5)!0.15cm!(E5) ); (P5_2) -- ( (P5_2)+(Q5_2)-(B5) ) -- (Q5_2); [align=center] at (0,-0.5) 2AE = 3DF , BE BH ; % (6) 筝形翻折 AB=AC,BD=CD [shift= (10,-3) , scale=1.5, x= (-0.3535cm,-0.3535cm) , y= (1cm,0cm) , z= (0cm,1cm) ] (O6) at (0,0,0); (B6) at (0,-1,0); (C6) at (0,1,0); (D6) at (1,0,0); (A6) at (0,0,1); (A6) -- node[midway, above left] a (B6); (A6) -- node[midway, above right] a (C6); (A6) -- (D6); (B6) -- node[midway, below left] b (D6); (C6) -- node[midway, below right] b (D6); [dashed] (B6) -- (C6); [dashed] (A6) -- (O6); [dashed] (D6) -- (O6); [above] at (A6) A ; [left] at (B6) B ; [right] at (C6) C ; [below] at (D6) D ; [above right] at (O6) O ; (P6) at ( (O6)!0.15cm!(A6) ); (Q6) at ( (O6)!0.15cm!(B6) ); (P6) -- ( (P6)+(Q6)-(O6) ) -- (Q6); (P6_2) at ( (O6)!0.15cm!(D6) ); (Q6_2) at ( (O6)!0.15cm!(C6) ); (P6_2) -- ( (P6_2)+(Q6_2)-(O6) ) -- (Q6_2); at (0, 0, -0.8) BC AD ;

几何法求距离和角度¶

定理 1. 对角线向量定理（空间余弦定理）¶

0.74 [ AC , BD = ( AD ^2 + BC ^2) - ( AB ^2 + CD ^2) 2 AC BD ] 说明：该定理既适用于平面向量（如四

题型 1. 等体积法¶

当直接计算某三棱锥体积不方便时，可考虑转换顶点来算体积. 在求点到平面的距离时，也可用等体积法. 例如，要求点 (A )到平面 (BCD )的距离 (d )，若

题型 2. 二面角的几何法¶

定义法：直接在两个半平面内找到与棱垂直的射线，它们的夹角即为二面角的平面角. 三垂线法：先过其中一个半平面的点作另一个半平面的垂线，再用三垂线定理作二面角的平面

定义法：直接在两个半平面内找到与棱垂直的射线，它们的夹角即为二面角的平面角. 三垂线法：先过其中一个半平面的点作另一个半平面的垂线，再用三垂线定理作二面角的平面角. 垂面法：作一与棱垂直的平面，该垂面与二面角两半平面相交，得到交线，交线所成的角为二面角的平面角. 关键在于找与二面角的棱垂直且与二面角两半平面都有交线的平面. 垂直转化：若已知平面 ( )内与棱 (l )垂直的直线 (a )和平面 ( )的垂线 (b )，则可用 (a )和 (b )求夹角，该夹角与二面角 ( - l- )互余（或与其补角互余）. 射影面积法：设 ( ABC ) 的面积为 (S )， ( ABC ) 在平面 ( ) 内的射影三角形为 ( A'B'C' )，其面积为 (S' )，则平面 (ABC ) 与平面 ( ) 所成锐二面角的余弦值 ( = S' S ). 0.19 [scale=0.9,line cap=round, line join=round] x= (0.35cm, -0.35cm) , y= (1cm, 0cm) , z= (0cm, 1cm) (0,0,0) -- (3,0,0) coordinate(L2) -- (3, 2.5, 0) coordinate(B2) -- (0, 2.5, 0) coordinate(B1) -- cycle; at (2.6, 2.2, 0) ; (A1) at (0, 1.5, 2.0); (A2) at (3, 1.5, 2.0); (0,0,0) -- (L2) -- (A2) -- (A1) -- cycle; at (0.4, 1.0, 1.4) ; [thick] (0,0,0) -- (L2); [below left] at (1,0,0) l ; (O) at (1.5, 0, 0); (P) at (1.5, 1.2, 1.6); (Q) at (1.5, 2.0, 0); [thick] (O) -- (P); [thick] (O) -- (Q); [dashed, gray] (P) -- (Q); (1.7, 0, 0) -- (1.7, 0.2, 0) -- (1.5, 0.2, 0); (1.2, 0, 0) -- (1.2, 0.15, 0.2) -- (1.5, 0.15, 0.2); (arc1) at (1.5, 0.3, 0.4); (arc2) at (1.5, 0.5, 0); (arc2) to[bend right=25] (arc1); at (1.5, 0.6, 0.3) ; % % 0.19 [scale=0.9, line cap=round, line join=round] x= (0.35cm, -0.35cm) , y= (1cm, 0cm) , z= (0cm, 1cm) (0,0,0) -- (3,0,0) coordinate(L2) -- (3, 2.5, 0) coordinate(B2) -- (0, 2.5, 0) coordinate(B1) -- cycle; at (2.6, 2.2, 0) ; (A1) at (0, 1.5, 2.0); (A2) at (3, 1.5, 2.0); (0,0,0) -- (L2) -- (A2) -- (A1) -- cycle; at (0.4, 1.0, 1.4) ; [thick] (0,0,0) -- (L2); [below left] at (1,0,0) l ; (P) at (1.5, 1.2, 1.6); (H) at (1.5, 1.2, 0); (O) at (1.5, 0, 0); [thick] (P) -- (O); [thick] (P) -- (H); [thick] (H) -- (O); [dashed, gray] (O) -- (1.5, 2.0, 0); (1.5, 1.0, 0) -- (1.5, 1.0, 0.2) -- (1.5, 1.2, 0.2); (1.7, 0, 0) -- (1.7, 0.2, 0) -- (1.5, 0.2, 0); (1.2, 0, 0) -- (1.2, 0.15, 0.2) -- (1.5, 0.15, 0.2); (arc1) at (1.5, 0.3, 0.4); (arc2) at (1.5, 0.5, 0); (arc2) to[bend right=25] (arc1); at (1.5, 0.6, 0.3) ; [above] at (P) P ; [right] at (H) A ; [below] at (O) O ; % % 0.19 [scale=0.9, line cap=round, line join=round] x= (0.35cm, -0.35cm) , y= (1cm, 0cm) , z= (0cm, 1cm) (0,0,0) -- (3,0,0) coordinate(L2) -- (3, 2.5, 0) coordinate(B2) -- (0, 2.5, 0) coordinate(B1) -- cycle; at (2.6, 2.2, 0) ; (A1) at (0, 1.5, 2.0); (A2) at (3, 1.5, 2.0); (0,0,0) -- (L2) -- (A2) -- (A1) -- cycle; at (0.4, 1.0, 1.4) ; [thick] (0,0,0) -- (L2); [below left] at (1,0,0) l ; [blue!10, opacity=0.5] (1.5, -0.5, -0.5) -- (1.5, 2.8, -0.5) -- (1.5, 2.8, 2.0) -- (1.5, -0.5, 2.0) -- cycle; [blue!60, thick] (1.5, -0.5, -0.5) -- (1.5, 2.8, -0.5) -- (1.5, 2.8, 2.0) -- (1.5, -0.5, 2.0) -- cycle; [blue, right] at (1.5, 2.4, 1.8) ; [thick, red] (1.5, 0, 0) -- (1.5, 2.5, 0); [thick, red] (1.5, 0, 0) -- (1.5, 1.2, 1.6); [red] (1.5, 0.5, 0) to[bend right=25] (1.5, 0.3, 0.4); [red] at (1.5, 0.6, 0.3) ; % % 0.19 [scale=0.9, line cap=round, line join=round] x= (0.35cm, -0.35cm) , y= (1cm, 0cm) , z= (0cm, 1cm) (0,0,0) -- (3,0,0) coordinate(L2) -- (3, 2.5, 0) coordinate(B2) -- (0, 2.5, 0) coordinate(B1) -- cycle; at (2.6, 2.2, 0) ; (A1) at (0, 1.5, 2.0); (A2) at (3, 1.5, 2.0); (0,0,0) -- (L2) -- (A2) -- (A1) -- cycle; at (0.4, 1.0, 1.4) ; [thick] (0,0,0) -- (L2); [below left] at (1,0,0) l ; (O) at (1.5, 0, 0); (P) at (1.5, 0.6, 0.8); [thick] (O) -- (1.5, 1.2, 1.6) node[right] a ; [thick] (P) -- (1.5, 0.6, 1.5) node[left] b ; [thick] (1.5, 0.6, 0) -- (P); [thick] (O) -- (1.5, 0.6, 0); (1.2, 0, 0) -- (1.2, 0.15, 0.2) -- (1.5, 0.15, 0.2); (arc1) at (1.5, 0.6, 1.1); (arc2) at (1.5, 0.8, 1); (arc1) to[bend left=20] (arc2); at (1.5, 0.8, 1.4) ; % % 0.19 [scale=0.9, line cap=round, line join=round] x= (0.35cm, -0.35cm) , y= (1cm, 0cm) , z= (0cm, 1cm) (0,0,0) -- (3,0,0) coordinate(L2) -- (3, 2.8, 0) coordinate(B2) -- (0, 2.8, 0) coordinate(B1) -- cycle; at (2.6, 2.5, 0) ; (A) at (0.5, 0.5, 0.6); (B) at (1.2, 2.2, 1.5); (C) at (2.5, 1, 1); (Ap) at (0.5, 0.5, 0); (Bp) at (1.2, 2.2, 0); (Cp) at (2.5, 1, 0); [thick] (A) -- (B) node[midway, above left] S -- (C) -- cycle; [blue, thick] (Ap) -- (Bp) node[midway, below right, black] S' -- (Cp) -- cycle; [dashed] (A) -- (Ap); [dashed] (B) -- (Bp); [dashed] (C) -- (Cp); [above] at (A) A ; [above] at (B) B ; [right] at (C) C ; [left] at (Ap) A' ; [right] at (Bp) B' ; [below] at (Cp) C' ; %

二级结论¶

结论 1. 四面体体积公式¶

已知异面直线段 ( AB = a )， ( CD = b )，异面直线夹角为 ( )，且异面直线的距离为 ( d )，则四面体 ( ABCD ) 的体积为： [

结论 2. 三棱锥与向量四心（人教A必修二P152）¶

定理 1. 三正弦定理（最大角定理）¶

0.64 直线 ( AB ) 平面 ( BCD )，直线 ( CD ) 平面 ( ABC )，则 ( ) 为直线 ( AD ) 与平面 ( ABC ) 所成线面

定理 2. 三余弦定理（最小角定理）¶

0.64 直线 ( AB ) 平面 ( BCD )，直线 ( CD ) 平面 ( ABC )，则 ( ) 为直线 ( AD ) 与平面 ( ABC ) 所成线面

定理 3. 祖暅原理（人教A必修二P121；人教B必修四P85）¶

祖暅（gèng）于 5 世纪末提出了下面的体积计算原理：“幂势既同，则积不容异”. 这就是 “祖暅原理”. “势” 即是高，“幂” 是面积，祖暅原理用现代语言可

祖暅（gèng）于 5 世纪末提出了下面的体积计算原理：“幂势既同，则积不容异”. 这就是 “祖暅原理”. “势” 即是高，“幂” 是面积，祖暅原理用现代语言可以描述为： 0.69 夹在两个平行平面之间的两个几何体，被平行于这两个平面的任意平面所截，如果截得的两个截面的面积总相等，那么这两个几何体的体积相等. 如图，夹在平行平面间的两个几何体（它们的形状可以不同），被平行于这两个平面的任何一个平面所截，如果截面（阴影部分）的面积都相等，那么这两个几何体的体积一定相等. 0.3 ! [yscale=0.9, >=stealth, thick, line join=round, line cap=round] % 2. 底部平面 [fill=cyan!25, draw=black] (-1.5, -0.3) -- (6.0, -0.3) -- (8.0, 1.7) -- (0.5, 1.7) -- cycle; % 左侧棱台坐标点 (B5) at (0.4, 0.9); (B1) at (0.8, 0.4); (B2) at (2.2, 0.3); (B3) at (3.0, 0.8); (B4) at (2.0, 1.4); (T5) at (1.05, 4.25); (T1) at (1.25, 4.0); (T2) at (1.95, 3.95); (T3) at (2.35, 4.2); (T4) at (1.85, 4.5); % 左侧利用插值锁定平级剖面 ( =0.55保证左右视觉等高切割) (M5) at ( (B5)!0.55!(T5) ); (M1) at ( (B1)!0.55!(T1) ); (M2) at ( (B2)!0.55!(T2) ); (M3) at ( (B3)!0.55!(T3) ); (M4) at ( (B4)!0.55!(T4) ); % ---------------- % 右侧曲面体(使用相似几何参数化，确保底、中、顶三个横截面严格数学相似) % ---------------- #1#2#3#4 (#1_L) at ( #2 - 1.1*#4 , #3 + 0.05*#4 ); (#1_C1) at ( #2 - 1.0*#4 , #3 - 0.5*#4 ); (#1_C2) at ( #2 + 0.6*#4 , #3 - 0.6*#4 ); (#1_R) at ( #2 + 1.1*#4 , #3 - 0.05*#4 ); (#1_C3) at ( #2 + 0.7*#4 , #3 + 0.8*#4 ); (#1_C4) at ( #2 - 0.9*#4 , #3 + 0.7*#4 ); % 底面 (缩放1.0) RB 5.1 0.8 1.0 % 顶面 (高度与左侧对齐，缩放0.5) RT 5.1 4.25 0.5 % 中间平级截面 ( =0.55进行严格仿射空间插值： y=0.45*0.8+0.55*4.25=2.6975, scale=0.45*1.0+0.55*0.5=0.725) RM 5.1 2.6975 0.725 % 不可见的隐藏虚线 % 左侧 [dashed] (B3)--(B4)--(B5); [dashed] (B4)--(T4); % 右侧 (底部背侧边界) [dashed] (RB_R) .. controls (RB_C3) and (RB_C4) .. (RB_L); % 棕色横截面 (前实后虚表现透视深度) % 左侧截面 [brown!60] (M1)--(M2)--(M3)--(M4)--(M5)--cycle; (M5)--(M1)--(M2)--(M3); [dashed] (M3)--(M4)--(M5); % 右侧截面 [brown!60] (RM_L) .. controls (RM_C1) and (RM_C2) .. (RM_R) .. controls (RM_C3) and (RM_C4) .. (RM_L); (RM_L) .. controls (RM_C1) and (RM_C2) .. (RM_R); [dashed] (RM_R) .. controls (RM_C3) and (RM_C4) .. (RM_L); % ---------------- % 正面可视化实体的半透明填色遮罩 (罩在截面之上模拟容积穿透感) % ---------------- % 左侧 [orange!25, fill opacity=0.6] (B5)--(B1)--(B2)--(B3)--(T3)--(T2)--(T1)--(T5)--cycle; % 右侧外轮廓封套 [orange!25, fill opacity=0.6] (RB_L) .. controls (RB_C1) and (RB_C2) .. (RB_R) -- (RT_R) .. controls (RT_C3) and (RT_C4) .. (RT_L) -- cycle; % 描绘遮罩外部的正面轮廓实线与表面投影接缝 % 左侧 (B5)--(B1)--(B2)--(B3); (B5)--(T5); (B1)--(T1); (B2)--(T2); (B3)--(T3); % 右侧 (RB_L) .. controls (RB_C1) and (RB_C2) .. (RB_R); (RB_L) -- (RT_L); (RB_R) -- (RT_R); % 1. 顶部平面 (45度侧仰，dx=dy=2.0) [fill=cyan!50, fill opacity=0.5, draw=black] (-1.5, 3.2) -- (6.0, 3.2) -- (8.0, 5.2) -- (0.5, 5.2) -- cycle; % 顶部截口 (使用白色填充覆盖上层浅蓝平面以形成打孔感，强化截面) % 左侧 [fill=white, draw=black] (T1)--(T2)--(T3)--(T4)--(T5)--cycle; % 右侧 [fill=white, draw=black] (RT_L) .. controls (RT_C1) and (RT_C2) .. (RT_R) .. controls (RT_C3) and (RT_C4) .. (RT_L); % 基于祖暅原理推导球体积：设半球半径为 ( R )，构造辅助体 ( )：底面半径 ( R )、高 ( R ) 的圆柱挖去同底等高圆锥，记半球为 ( )，两几何体均夹于平面 ( z = 0 )（底面）与 ( z = R )（半球顶/圆柱上底）间. 取任意平面 ( z = h )（ ( 0 h R ) ）截两几何体：半球轴截面为半圆，由勾股定理，截面圆半径 ( r_1 = R^2 - h^2 )，截面面积 ( S_ (h) = r_1^2 = (R^2 - h^2) ) . 圆柱截面面积 ( R^2 )，圆锥截面因相似性（高与半径比 ( 1:1 ) ），半径 ( r_2 = h )，截面面积 ( h^2 )，故 ( ) 截面面积 ( S_ (h) = R^2 - h^2 = (R^2 - h^2) ) . 由于 ( S_ (h) = S_ (h) ) 对任意 ( h [0,R] ) 成立. 故 ( V_ = V_ ) . 半球体积 (V_ 半球 = V_ = V_ 圆柱 - V_ 圆锥 = R^3 - 1 3 R^3 = 2 3 R^3 ) ，从而球体积 ( V_ 球 = 4 3 R^3 ). % % 0.8 % % [trim=0 0 200pt 0, width= ] 祖暅原理球.png % % [>=stealth, line join=round, line cap=round] 1.1 x = (-0.75* cm,-0.15* cm) , y = (0.75* cm,-0.15* cm) , z = (0cm, cm) , 2.5 1.5 135 315 % ========== 左侧图：圆柱挖去圆锥 ========== [xshift=-4cm] (O) at (0,0,0); (O_top) at (0,0, ); (O_mid) at (0,0, ); (BL) at ( cos( ) , sin( ) , 0); (TL) at ( cos( ) , sin( ) , ); (BR) at ( cos( ) , sin( ) , 0); (TR) at ( cos( ) , sin( ) , ); (P_inner) at ( cos( ) , sin( ) , ); (P_outer) at ( cos( ) , sin( ) , ); % 1. 截面底层填充 (引入even odd rule表示空心圆环) [blue!80!black, opacity=0.3, even odd rule] plot[domain=0:360, samples=60] ( cos( ) , sin( ) , ) plot[domain=0:360, samples=60] ( cos( ) , sin( ) , ); % 2. 可见实线部分（底层背景、截面背面边线） plot[domain=315:495, samples=30] ( cos( ) , sin( ) , 0); plot[domain=315:495, samples=30] ( cos( ) , sin( ) , ); plot[domain=315:495, samples=30] ( cos( ) , sin( ) , ); [dashed] (O) -- (O_top); % 3. 不可见虚线部分（圆柱边缘、底面和截面的可视前缘） [dashed] plot[domain=135:315, samples=30] ( cos( ) , sin( ) , 0); [dashed] plot[domain=135:315, samples=30] ( cos( ) , sin( ) , ); [dashed] plot[domain=135:315, samples=30] ( cos( ) , sin( ) , ); plot[domain=0:360, samples=60] ( cos( ) , sin( ) , ); (BL) -- (TL); (BR) -- (TR); % 4. 圆锥内壁边界（俯视凹洞，内壁母线完全可见故作实线） (O) -- (TL); (O) -- (TR); % 5. 特殊高亮与端点 [dashed] (O_mid) -- (P_inner); [thick, blue] (P_inner) -- (P_outer); [fill=black] (O_mid) circle (1.5pt); [fill=blue, draw=blue] (P_inner) circle (1.5pt); [fill=blue, draw=blue] (P_outer) circle (1.5pt); [fill=black] (O) circle (1.5pt); [fill=black] (O_top) circle (1.5pt); % 6. 标注文本 [below] at ( (O_mid)!0.5!(P_inner) ) h ; [below] at ( (P_inner)!0.5!(P_outer) ) R-h ; % 7. 辅助高度尺寸线 (BL) -- ++(-0.5,0.5,0) coordinate (DimB); (TL) -- ++(-0.5,0.5,0) coordinate (DimT); [<->] (DimB) -- (DimT) node[midway, fill=white, inner sep=1pt] R ; (DimL_B) at ( cos( ) , sin( ) , 0); (DimL_T) at ( cos( ) , sin( ) , ); (DimL_T) -- ++(-0.3,0.3,0) coordinate (DimL_h); (DimL_B) -- ++(-0.3,0.3,0) coordinate (DimB_h); [<->] (DimL_h) -- (DimB_h) node[midway, fill=white, inner sep=1pt] h ; % ========== 右侧图：圆柱内接半球 ========== [xshift=4cm] sqrt( - ) (O) at (0,0,0); (O_top) at (0,0, ); (O_mid) at (0,0, ); (BL) at ( cos( ) , sin( ) , 0); (TL) at ( cos( ) , sin( ) , ); (BR) at ( cos( ) , sin( ) , 0); (TR) at ( cos( ) , sin( ) , ); (P_sec) at ( cos( ) , sin( ) , ); % 1. 截面底层填充 [blue!80!black, opacity=0.3] plot[domain=0:360, samples=60] ( cos( ) , sin( ) , ) -- cycle; % 2. 不可见虚线部分（底层背景、截面背面边线） plot[domain=315:495, samples=30] ( cos( ) , sin( ) , 0); plot[domain=315:495, samples=30] ( cos( ) , sin( ) , ); [dashed] (O) -- (O_top); % 3. 可见实线部分（外包圆柱与截面前缘） [dashed] plot[domain=135:315, samples=30] ( cos( ) , sin( ) , 0); [dashed] plot[domain=135:315, samples=30] ( cos( ) , sin( ) , ); plot[domain=0:360, samples=60] ( cos( ) , sin( ) , ); (BL) -- (TL); (BR) -- (TR); % 4. 【核心结构】以正等测透视特性的3D球面参数方程勾勒半球的完美抛物线轮廓特征 % 该方程提取自平行于投影视平面的球面大圆，画出来即是完美的三维球体表层轮廓半圆！ [thick] plot[domain=180:0, samples=50] ( cos( )/sqrt(2) - sin( )/sqrt(6) , - cos( )/sqrt(2) - sin( )/sqrt(6) , sin( )*2.15/sqrt(6) ); % 5. 特殊高亮与端点 [thick, blue] (O_mid) -- (P_sec); [fill=blue, draw=blue] (O_mid) circle (1.5pt); [fill=blue, draw=blue] (P_sec) circle (1.5pt); [fill=black] (O) circle (1.5pt); [fill=black] (O_top) circle (1.5pt); % 6. 标注文本 [below] at ( (O_mid)!0.5!(P_sec) ) R^2-h^2 ; % 7. R 尺寸线标注 (BR) -- ++(0.3,-0.3,0) coordinate (DimR_B); (TR) -- ++(0.3,-0.3,0) coordinate (DimR_T); [<->] (DimR_B) -- (DimR_T) node[midway, fill=white, inner sep=1pt] R ; [<->] (O) -- (BR) node[midway, fill=white, inner sep=1pt] R ;