Question 29
The diagonals of a convex quadrilateral $ABCD$ intersect at $P$. Let $L, M$ and $N$ be the points on the segments $AB, BC$ and $CD$, respectively, such that
$$
\frac{|AL|}{|LB|} = \frac{|BM|}{|MC|} = \frac{|CN|}{|ND|}.
$$
Suppose that the line $LM$ is tangent to the circumcircle of the triangle $MCN$ and the line $NM$ is tangent to the circumcircle of the triangle $MBL$. Prove that $\angle DPA = \angle NML$.
Illustrative figure.
$$
\frac{|AL|}{|LB|} = \frac{|BM|}{|MC|} = \frac{|CN|}{|ND|}.
$$
Suppose that the line $LM$ is tangent to the circumcircle of the triangle $MCN$ and the line $NM$ is tangent to the circumcircle of the triangle $MBL$. Prove that $\angle DPA = \angle NML$.
Illustrative figure.