已知任意凸四边形四个顶点坐标(x1,y1)---(x4,y4),求这个四边形2条对角线交点坐标(x,y)

Question

已知任意凸四边形四个顶点坐标(x1,y1)---(x4,y4),求这个四边形2条对角线交点坐标(x,y)

Answer 1

A(x1,y1) B(x2,y2) C(x3,y3) D(x4,y4)
对角线AC：y-y1=(y3-y1)/(x3-x1)*(x-x1)
对角线BD：y-y2=(y4-y2)/(x4-x2)*(x-x2)
(y3-y1)/(x3-x1)*(x-x1)+y1=(y4-y2)/(x4-x2)*(x-x2)+y2
(y3-y1)(x4-x2)(x-x1)+y1(x3-x1)(x4-x2)=(y4-y2)(x3-x1)(x-x2)+y2(x3-x1)(x4-x2)
(y3-y1)(x4-x2)(x-x1)-(y4-y2)(x3-x1)(x-x2)=(x3-x1)(x4-x2)(y2-y1)
x[(y3-y1)(x4-x2)-(y4-y2)(x3-x1)]=(x3-x1)(x4-x2)(y2-y1)+x1(y3-y1)(x4-x2)-x2(y4-y2)(x3-x1)
x=[(x3-x1)(x4-x2)(y2-y1)+x1(y3-y1)(x4-x2)-x2(y4-y2)(x3-x1)]/[(y3-y1)(x4-x2)-(y4-y2)(x3-x1)]
y=(y3-y1)/(x3-x1)*(x-x1)+y1
=(y3-y1)[(x4-x2)(y2-y1)+(x1-x2)(y4-y2)]/[(y3-y1)(x4-x2)-(y4-y2)(x3-x1)]+y1
所以交点坐标为
([(x3-x1)(x4-x2)(y2-y1)+x1(y3-y1)(x4-x2)-x2(y4-y2)(x3-x1)]/[(y3-y1)(x4-x2)-(y4-y2)(x3-x1)],(y3-y1)[(x4-x2)(y2-y1)+(x1-x2)(y4-y2)]/[(y3-y1)(x4-x2)-(y4-y2)(x3-x1)]+y1)