Question

If $a,b,c$ and $u,v,w$ are the complex numbers representing the vertices of two triangles such that $c=(1-r)a+rb$ and $w=(1-r)u+rv$, where $r$ is a complex number, then prove that the two triangles are similar.

Answer 1

Since $a,b,c$ and $u,v,w$ are the vertices of two triangles

Also, $c=(1-r)a+rb$ and $w=(1-r)u+rv$ .......$\left(1\right)$

Applying $R_3\to R_3-(1-r)R_1+r{R}_2$

$=\left|\begin{array}{ccc}a& u& 1\\ b& v& 1\\ c-(1-r)a-rb& w-(1-r)u-rv& 1-(1-r)-r\end{array}\right|$

$=\left|\begin{array}{ccc}a& u& 1\\ b& v& 1\\ 0& 0& 0\end{array}\right|$

$=0$

Hence, two triangles are similar.