Consider a convex pentagon such that Let be the point of intersection of the lines and . Prove that the line passes through the midpoint of the side .
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Consider a convex pentagon $ABCDE$ such that
$$\angle BAC = \angle CAD = \angle DAE\ \ \ ,\ \ \ \angle ABC = \angle ACD = \angle ADE$$
Let $P$ be the point of intersection of the lines $BD$ and $CE$. Prove that the line $AP$ passes through the midpoint of the side $CD$.
Source: Međunarodna matematička olimpijada, shortlist 2006
Školjka 
such that 
be the point of intersection of the lines 
and 
. Prove that the line 
passes through the midpoint of the side 
.