MathNet 2014 Question 25

Let $AB\Gamma$ an acute angled scalene triangle with $AB \lt A\Gamma \lt B\Gamma$. Let $\Delta, E, Z$ be the midpoints of the sides $B\Gamma, A\Gamma, AB$, respectively, and let $BK, G\Lambda$ be altitudes. At the extension of $\Delta Z$, to the part of $Z$, we consider a point $M$, such that the parallel from $M$ to $K\Lambda$ intersect the extensions of $\Gamma A, BA$ and $\Delta E$ at points $\Sigma, T$ and $N$, respectively. If the circumcircle of the triangle $MB\Delta$, say $(c_1)$, intersects the line $\Delta N$ at point $P$ and the circumcircle of the triangle $N\Gamma\Delta$, say $(c_2)$, intersects the line $\Delta M$ at point $\Pi$, prove that: $\Sigma T \parallel \Pi P$. (E. Psychas)

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