test-tikz

Problem Statement

Let $\triangle ABC$ be an acute-angled triangle with incircle $\omega$ touching sides $BC, CA, AB$ at points $D, E, F$ respectively. Let $P$ be the midpoint of altitude $AD$.

Prove that the line $DP$ intersects the incircle $\omega$ again at a point $Q$ such that the tangency lines at $E$ and $F$ intersect on the circumcircle of $\triangle ABC$.

Consider triangle $\triangle ABC$ with incircle $\omega$ centered at $I$ with radius $r$.

Key Lemma

By inversion centered at $D$ with radius $r = \sqrt{BD \cdot CD}$:

$$ \frac{AE}{EC} \cdot \frac{CD}{DB} \cdot \frac{BF}{FA} = 1 $$

Thus by Ceva's Theorem, the lines $AD, BE, CF$ are concurrent at the Gergonne point $G$.

$$ \implies \boxed{\angle FQE = \angle FDE = 90^\circ - \frac{\angle A}{2}} \quad \blacksquare $$