Incentre and excentre of a triangle

Question

Prove that the triangle formed by the points of contact of the sides of a given triangle with the excircles corresponding to these sides is equivalent to the triangle formed by the points of contact of the sides of the triangle with the inscribed circle

Answer 1

Hint: To show the two triangles are equivalent (or having the same area), note that $$area(\triangle AEF)=\frac12(p-a)^2\sin A =\frac 12(p-a)^2\frac{a}{2R} = \frac{(p-a)^2(p-b+p-c)}{4R}$$ where $a,b,c$ are the sides of $\triangle ABC$ and $p=\frac{a+b+c}{2}$.

Similarly for the triangles $\triangle AE'E',\dots$