A straight line is drawn through the incenter $I$ of $\Delta ABC$...

Question

A straight line is drawn through the incenter $I$ of $\Delta ABC, $ perpendicular to $AI$ meeting $AB,AC $ in $D$ and $E$ respectively. Prove that $BD\cdot CE=ID^2.$

I am unable to solve the above problem.

Any help will be appreciated.

Answer 1

1) Will $\angle TID = 0.5 \angle A$?

2) Will $x = 0.5 \angle C$?

3) Will the red marked angles equal?

If the answers to the above are all yes, then the red triangle is similar to the green. By noting that DI =IE, we obtain the said equality.