Let $ABC$ be a triangle, $P$, $Q$ be two isogonal conjugate. $AP$, $AQ$ meets (ABC) at $D, E$ respectively. Two lines through $D, E$ meet (ABC) at $T, N$ and meet BC at $G, H$ respectively. Let $PG, HQ$ meets $(GHNT)$ again at $K, F$. Then $K, F, A$ are collinear. enter image description here


1 Answer 1


$\textbf{Proof :}$

let $P$ moves on $AD$ with constant speed, so then $Q$ moves on $AE$ with projectively constant speed.

$D, E, T, N, G, H$ are fixed, so $F, K$ moves on fixed $(GHTN)$ with projectively constant speed. And so we need to prove that $A, K, F$ are collinear for some $3$ different cases of $P\in AD$.

Just take $P = A, P= AD\cap BC, P\in (ABC)\cap AD$. $\Box$


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