Question

第一象限内有等腰三角形,已知顶点坐标,底边中点的坐标和底边长,求另两点坐标.

Answer 1

复数法：

设复数DA为z=a+bi=r(cosθ+isinθ)

其中a=Ax-Dx b=Ay-Dy

则复数DB为DA以D为中心逆向旋转π/2

DB

=z(cosπ/2+isinπ/2)

=r[cos(θ+π/2)+isin(θ+π/2)]

=r[-sinθ+icosθ]

=-b+ai

复数DC为DA以D为中心正向旋转π/2

DC

=z(cos-π/2+isin-π/2)

=r[cos(θ-π/2)+isin(θ-π/2)]

=r[sinθ+icosθ]

=b-ai

∴DB=Dy-Ay+(Ax-Dx)i

DC=Ay-Dy+(Dx-Ax)i

∴复数B

=复数DB+复数D

=Dy-Ay+(Ax-Dx)i+Dx+Dyi

=Dx+Dy-Ay+(Ax-Dx+Dy)i

复数C

=复数DC+复数D

=Ay-Dy+(Dx-Ax)i+Dx+Dyi

=Dx-Dy+Ay+(Dx-Ax+Dy)i

坐标：

B(Dx+Dy-Ay,Ax-Dx+Dy)

C(Dx-Dy+Ay,Dx-Ax+Dy)

∴

Bx=Dx+Dy-Ay

By=Ax-Dx+Dy

Cx=Dx-Dy+Ay

Cy=Dx-Ax+Dy