三角型的性质
2015-08-17 14:14
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三角形顶点坐标:A(x1; y1);B(x2; y2);C(x3; y3);
④重心G(x4;y4);x4=(x1+x2+x3)/3;y4=(y1+y2+y3)/3;
⑤外心W(x5;y5);根据外心到各顶点的距离相等:AG=BG;AG=CG;即:
Sqrt[(x1 - x5)^2 + (y1 - y5)^2] == Sqrt[(x2 - x5)^2 + (y2 - y5)^2],
Sqrt[(x1 - x5)^2 + (y1 - y5)^2] == Sqrt[(x3 - x5)^2 + (y3 - y5)^2]
解得:x5 = (x2^2 y1 - x3^2 y1 - x1^2 y2 + x3^2 y2 - y1^2 y2 + y1 y2^2 + x1^2 y3 - x2^2 y3 + y1^2 y3 - y2^2 y3 - y1 y3^2 + y2 y3^2)/(2 (x2 y1 - x3 y1 - x1 y2 + x3 y2 + x1 y3 - x2 y3));
y5 = -(-x1^2 x2 + x1 x2^2 + x1^2 x3 - x2^2 x3 - x1 x3^2 + x2 x3^2 - x2 y1^2 + x3 y1^2 + x1 y2^2 - x3 y2^2 - x1 y3^2 + x2 y3^2)/(2 (x2 y1 - x3 y1 - x1 y2 + x3 y2 + x1 y3 - x2 y3));
⑥内心N(x6;y6);根据内心到各边的距离相等:先求内心到各边垂线垂足与顶点的距离;
1/2 (Sqrt[(x1 - x2)^2 + (y1 - y2)^2] + Sqrt[(x1 - x3)^2 + (y1 - y3)^2] - Sqrt[(x2 - x3)^2 + (y2 - y3)^2]);
1/2 (Sqrt[(x1 - x2)^2 + (y1 - y2)^2] - Sqrt[(x1 - x3)^2 + (y1 - y3)^2] + Sqrt[(x2 - x3)^2 + (y2 - y3)^2]);
1/2 (-Sqrt[(x1 - x2)^2 + (y1 - y2)^2] + Sqrt[(x1 - x3)^2 + (y1 - y3)^2] + Sqrt[(x2 - x3)^2 + (y2 - y3)^2]);
计算内心到个顶点的距离;根据勾股定理计算内心到各边的距离,根据距离相等列方程:
(x1 - x6)^2 - 1/4 (Sqrt[(x1 - x2)^2 + (y1 - y2)^2] + Sqrt[(x1 - x3)^2 + (y1 - y3)^2] - Sqrt[(x2 - x3)^2 + (y2 - y3)^2])^2 + (y1 - y6)^2 == (x2 - x6)^2 - 1/4 (Sqrt[(x1 - x2)^2 + (y1 - y2)^2] - Sqrt[(x1 - x3)^2 + (y1 - y3)^2] + Sqrt[(x2 - x3)^2 + (y2 - y3)^2])^2
+ (y2 - y6)^2,(x1 - x6)^2 - 1/4 (Sqrt[(x1 - x2)^2 + (y1 - y2)^2] + Sqrt[(x1 - x3)^2 + (y1 - y3)^2] - Sqrt[(x2 - x3)^2 + (y2 - y3)^2])^2 + (y1 - y6)^2 == (x3 - x6)^2 - 1/4 (-Sqrt[(x1 - x2)^2 + (y1 - y2)^2] + Sqrt[(x1 - x3)^2 + (y1 - y3)^2] + Sqrt[(x2 - x3)^2
+ (y2 - y3)^2])^2 + (y3 - y6)^2
解得:x6 = (x2^2 y1 - x3^2 y1 - x1^2 y2 + x3^2 y2 - y1^2 y2 + y1 y2^2 + x1^2 y3 - x2^2 y3 + y1^2 y3 - y2^2 y3 - y1 y3^2 + y2 y3^2 + y2 Sqrt[x1^2 - 2 x1 x2 + x2^2 + y1^2 - 2 y1 y2 + y2^2] Sqrt[x1^2 - 2 x1 x3 + x3^2 + y1^2 - 2 y1 y3 + y3^2] - Sqrt[x1^2 - 2 x1 x2
+ x2^2 + y1^2 - 2 y1 y2 + y2^2] y3 Sqrt[x1^2 - 2 x1 x3 + x3^2 + y1^2 - 2 y1 y3 + y3^2] - y1 Sqrt[x1^2 - 2 x1 x2 + x2^2 + y1^2 - 2 y1 y2 + y2^2] Sqrt[x2^2 - 2 x2 x3 + x3^2 + y2^2 - 2 y2 y3 + y3^2] + Sqrt[x1^2 - 2 x1 x2 + x2^2 + y1^2 - 2 y1 y2 + y2^2] y3 Sqrt[x2^2
- 2 x2 x3 + x3^2 + y2^2 - 2 y2 y3 + y3^2] + y1 Sqrt[x1^2 - 2 x1 x3 + x3^2 + y1^2 - 2 y1 y3 + y3^2] Sqrt[x2^2 - 2 x2 x3 + x3^2 + y2^2 - 2 y2 y3 + y3^2] - y2 Sqrt[x1^2 - 2 x1 x3 + x3^2 + y1^2 - 2 y1 y3 + y3^2] Sqrt[x2^2 - 2 x2 x3 + x3^2 + y2^2 - 2 y2 y3 + y3^2])/(2
(x2 y1 - x3 y1 - x1 y2 + x3 y2 + x1 y3 - x2 y3));
y6 = -(-x1^2 x2 + x1 x2^2 + x1^2 x3 - x2^2 x3 - x1 x3^2 + x2 x3^2 - x2 y1^2 + x3 y1^2 + x1 y2^2 - x3 y2^2 - x1 y3^2 + x2 y3^2 + x2 Sqrt[x1^2 - 2 x1 x2 + x2^2 + y1^2 - 2 y1 y2 + y2^2] Sqrt[x1^2 - 2 x1 x3 + x3^2 + y1^2 - 2 y1 y3 + y3^2] - x3 Sqrt[x1^2 - 2 x1
x2 + x2^2 + y1^2 - 2 y1 y2 + y2^2] Sqrt[x1^2 - 2 x1 x3 + x3^2 + y1^2 - 2 y1 y3 + y3^2] - x1 Sqrt[x1^2 - 2 x1 x2 + x2^2 + y1^2 - 2 y1 y2 + y2^2] Sqrt[x2^2 - 2 x2 x3 + x3^2 + y2^2 - 2 y2 y3 + y3^2] + x3 Sqrt[x1^2 - 2 x1 x2 + x2^2 + y1^2 - 2 y1 y2 + y2^2] Sqrt[x2^2
- 2 x2 x3 + x3^2 + y2^2 - 2 y2 y3 + y3^2] + x1 Sqrt[x1^2 - 2 x1 x3 + x3^2 + y1^2 - 2 y1 y3 + y3^2] Sqrt[x2^2 - 2 x2 x3 + x3^2 + y2^2 - 2 y2 y3 + y3^2] - x2 Sqrt[x1^2 - 2 x1 x3 + x3^2 + y1^2 - 2 y1 y3 + y3^2] Sqrt[x2^2 - 2 x2 x3 + x3^2 + y2^2 - 2 y2 y3 + y3^2])/(2
(x2 y1 - x3 y1 - x1 y2 + x3 y2 + x1 y3 - x2 y3));
⑦垂心H(x7;y7);分别做高线:
AH⊥BC;BH⊥AC;(y1 - y7)/(x1 - x7) (y2 - y3)/(x2 - x3) == -1,(y2 - y7)/(x2 - x7) (y1 - y3)/(x1 - x3) == -1
解得:x7 = -(x1 x2 y1 - x1 x3 y1 - x1 x2 y2 + x2 x3 y2 + y1^2 y2 - y1 y2^2 + x1 x3 y3 - x2 x3 y3 - y1^2 y3 + y2^2 y3 + y1 y3^2 - y2 y3^2)/(-x2 y1 + x3 y1 + x1 y2 - x3 y2 - x1 y3 + x2 y3);
y7 = -(x1^2 x2 - x1 x2^2 - x1^2 x3 + x2^2 x3 + x1 x3^2 - x2 x3^2 + x1 y1 y2 - x2 y1 y2 - x1 y1 y3 + x3 y1 y3 + x2 y2 y3 - x3 y2 y3)/(x2 y1 - x3 y1 - x1 y2 + x3 y2 + x1 y3 - x2 y3);
外接圆圆心
a=((x1*x1+y1*y1-x2*x2-y2*y2)*(y1-y3)-(x1*x1+y1*y1-x3*x3-y3*y3)*(y1-y2))/(2.0*((y1-y3)*(x1-x2)-(y1-y2)*(x1-x3)));
b=((x1*x1+y1*y1-x2*x2-y2*y2)*(x1-x3)-(x1*x1+y1*y1-x3*x3-y3*y3)*(x1-x2))/(2.0*((x1-x3)*(y1-y2)-(x1-x2)*(y1-y3)));
④重心G(x4;y4);x4=(x1+x2+x3)/3;y4=(y1+y2+y3)/3;
⑤外心W(x5;y5);根据外心到各顶点的距离相等:AG=BG;AG=CG;即:
Sqrt[(x1 - x5)^2 + (y1 - y5)^2] == Sqrt[(x2 - x5)^2 + (y2 - y5)^2],
Sqrt[(x1 - x5)^2 + (y1 - y5)^2] == Sqrt[(x3 - x5)^2 + (y3 - y5)^2]
解得:x5 = (x2^2 y1 - x3^2 y1 - x1^2 y2 + x3^2 y2 - y1^2 y2 + y1 y2^2 + x1^2 y3 - x2^2 y3 + y1^2 y3 - y2^2 y3 - y1 y3^2 + y2 y3^2)/(2 (x2 y1 - x3 y1 - x1 y2 + x3 y2 + x1 y3 - x2 y3));
y5 = -(-x1^2 x2 + x1 x2^2 + x1^2 x3 - x2^2 x3 - x1 x3^2 + x2 x3^2 - x2 y1^2 + x3 y1^2 + x1 y2^2 - x3 y2^2 - x1 y3^2 + x2 y3^2)/(2 (x2 y1 - x3 y1 - x1 y2 + x3 y2 + x1 y3 - x2 y3));
⑥内心N(x6;y6);根据内心到各边的距离相等:先求内心到各边垂线垂足与顶点的距离;
1/2 (Sqrt[(x1 - x2)^2 + (y1 - y2)^2] + Sqrt[(x1 - x3)^2 + (y1 - y3)^2] - Sqrt[(x2 - x3)^2 + (y2 - y3)^2]);
1/2 (Sqrt[(x1 - x2)^2 + (y1 - y2)^2] - Sqrt[(x1 - x3)^2 + (y1 - y3)^2] + Sqrt[(x2 - x3)^2 + (y2 - y3)^2]);
1/2 (-Sqrt[(x1 - x2)^2 + (y1 - y2)^2] + Sqrt[(x1 - x3)^2 + (y1 - y3)^2] + Sqrt[(x2 - x3)^2 + (y2 - y3)^2]);
计算内心到个顶点的距离;根据勾股定理计算内心到各边的距离,根据距离相等列方程:
(x1 - x6)^2 - 1/4 (Sqrt[(x1 - x2)^2 + (y1 - y2)^2] + Sqrt[(x1 - x3)^2 + (y1 - y3)^2] - Sqrt[(x2 - x3)^2 + (y2 - y3)^2])^2 + (y1 - y6)^2 == (x2 - x6)^2 - 1/4 (Sqrt[(x1 - x2)^2 + (y1 - y2)^2] - Sqrt[(x1 - x3)^2 + (y1 - y3)^2] + Sqrt[(x2 - x3)^2 + (y2 - y3)^2])^2
+ (y2 - y6)^2,(x1 - x6)^2 - 1/4 (Sqrt[(x1 - x2)^2 + (y1 - y2)^2] + Sqrt[(x1 - x3)^2 + (y1 - y3)^2] - Sqrt[(x2 - x3)^2 + (y2 - y3)^2])^2 + (y1 - y6)^2 == (x3 - x6)^2 - 1/4 (-Sqrt[(x1 - x2)^2 + (y1 - y2)^2] + Sqrt[(x1 - x3)^2 + (y1 - y3)^2] + Sqrt[(x2 - x3)^2
+ (y2 - y3)^2])^2 + (y3 - y6)^2
解得:x6 = (x2^2 y1 - x3^2 y1 - x1^2 y2 + x3^2 y2 - y1^2 y2 + y1 y2^2 + x1^2 y3 - x2^2 y3 + y1^2 y3 - y2^2 y3 - y1 y3^2 + y2 y3^2 + y2 Sqrt[x1^2 - 2 x1 x2 + x2^2 + y1^2 - 2 y1 y2 + y2^2] Sqrt[x1^2 - 2 x1 x3 + x3^2 + y1^2 - 2 y1 y3 + y3^2] - Sqrt[x1^2 - 2 x1 x2
+ x2^2 + y1^2 - 2 y1 y2 + y2^2] y3 Sqrt[x1^2 - 2 x1 x3 + x3^2 + y1^2 - 2 y1 y3 + y3^2] - y1 Sqrt[x1^2 - 2 x1 x2 + x2^2 + y1^2 - 2 y1 y2 + y2^2] Sqrt[x2^2 - 2 x2 x3 + x3^2 + y2^2 - 2 y2 y3 + y3^2] + Sqrt[x1^2 - 2 x1 x2 + x2^2 + y1^2 - 2 y1 y2 + y2^2] y3 Sqrt[x2^2
- 2 x2 x3 + x3^2 + y2^2 - 2 y2 y3 + y3^2] + y1 Sqrt[x1^2 - 2 x1 x3 + x3^2 + y1^2 - 2 y1 y3 + y3^2] Sqrt[x2^2 - 2 x2 x3 + x3^2 + y2^2 - 2 y2 y3 + y3^2] - y2 Sqrt[x1^2 - 2 x1 x3 + x3^2 + y1^2 - 2 y1 y3 + y3^2] Sqrt[x2^2 - 2 x2 x3 + x3^2 + y2^2 - 2 y2 y3 + y3^2])/(2
(x2 y1 - x3 y1 - x1 y2 + x3 y2 + x1 y3 - x2 y3));
y6 = -(-x1^2 x2 + x1 x2^2 + x1^2 x3 - x2^2 x3 - x1 x3^2 + x2 x3^2 - x2 y1^2 + x3 y1^2 + x1 y2^2 - x3 y2^2 - x1 y3^2 + x2 y3^2 + x2 Sqrt[x1^2 - 2 x1 x2 + x2^2 + y1^2 - 2 y1 y2 + y2^2] Sqrt[x1^2 - 2 x1 x3 + x3^2 + y1^2 - 2 y1 y3 + y3^2] - x3 Sqrt[x1^2 - 2 x1
x2 + x2^2 + y1^2 - 2 y1 y2 + y2^2] Sqrt[x1^2 - 2 x1 x3 + x3^2 + y1^2 - 2 y1 y3 + y3^2] - x1 Sqrt[x1^2 - 2 x1 x2 + x2^2 + y1^2 - 2 y1 y2 + y2^2] Sqrt[x2^2 - 2 x2 x3 + x3^2 + y2^2 - 2 y2 y3 + y3^2] + x3 Sqrt[x1^2 - 2 x1 x2 + x2^2 + y1^2 - 2 y1 y2 + y2^2] Sqrt[x2^2
- 2 x2 x3 + x3^2 + y2^2 - 2 y2 y3 + y3^2] + x1 Sqrt[x1^2 - 2 x1 x3 + x3^2 + y1^2 - 2 y1 y3 + y3^2] Sqrt[x2^2 - 2 x2 x3 + x3^2 + y2^2 - 2 y2 y3 + y3^2] - x2 Sqrt[x1^2 - 2 x1 x3 + x3^2 + y1^2 - 2 y1 y3 + y3^2] Sqrt[x2^2 - 2 x2 x3 + x3^2 + y2^2 - 2 y2 y3 + y3^2])/(2
(x2 y1 - x3 y1 - x1 y2 + x3 y2 + x1 y3 - x2 y3));
⑦垂心H(x7;y7);分别做高线:
AH⊥BC;BH⊥AC;(y1 - y7)/(x1 - x7) (y2 - y3)/(x2 - x3) == -1,(y2 - y7)/(x2 - x7) (y1 - y3)/(x1 - x3) == -1
解得:x7 = -(x1 x2 y1 - x1 x3 y1 - x1 x2 y2 + x2 x3 y2 + y1^2 y2 - y1 y2^2 + x1 x3 y3 - x2 x3 y3 - y1^2 y3 + y2^2 y3 + y1 y3^2 - y2 y3^2)/(-x2 y1 + x3 y1 + x1 y2 - x3 y2 - x1 y3 + x2 y3);
y7 = -(x1^2 x2 - x1 x2^2 - x1^2 x3 + x2^2 x3 + x1 x3^2 - x2 x3^2 + x1 y1 y2 - x2 y1 y2 - x1 y1 y3 + x3 y1 y3 + x2 y2 y3 - x3 y2 y3)/(x2 y1 - x3 y1 - x1 y2 + x3 y2 + x1 y3 - x2 y3);
外接圆圆心
a=((x1*x1+y1*y1-x2*x2-y2*y2)*(y1-y3)-(x1*x1+y1*y1-x3*x3-y3*y3)*(y1-y2))/(2.0*((y1-y3)*(x1-x2)-(y1-y2)*(x1-x3)));
b=((x1*x1+y1*y1-x2*x2-y2*y2)*(x1-x3)-(x1*x1+y1*y1-x3*x3-y3*y3)*(x1-x2))/(2.0*((x1-x3)*(y1-y2)-(x1-x2)*(y1-y3)));
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