Problem (2013BalkanMOp1): In a triangle $ABC$, the excircle $\omega_a$ opposite $A$ touches $AB$ at $P$ and $AC$ at $Q$, while the excircle $\omega_b$ opposite $B$ touches $BA$ at $M$ and $BC$ at $N$. Let $K$ be the projection of $C$ onto $MN$ and let $L$ be the projection of $C$ onto $PQ$. Show that the quadrilateral $MKLP$ is cyclic.
AlphaGeometry formalization: a b c = triangle; i1 = excenter a b c; p = foot i1 a b; q = foot i1 a c; i2 = excenter b c a; m = foot i2 a b; n = foot i2 b c; k = foot c m n; l = foot c p q ? cyclic m k l p
HAGeo formalization: A B C = triangle; I1 = excenter A B C; I2 = excenter B A C; AB = line A B; P = foot I1 AB; AC = line A C; Q = foot I1 AC; M = foot I2 AB; BC = line B C; N = foot I2 BC; MN = line M N; K = foot C MN; PQ = line P Q; L = foot C PQ; Prove: concyclic M K L P
Difficulty score: 2.5