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Geometrical equality

Source: Mediterranean math competition 2018

June 6, 2018
geometrycircumcircleUgly

Problem Statement

Let ABCABC be acute triangle. Let EE and FF be points on BCBC, such that angles BAEBAE and FACFAC are equal. Lines AEAE and AFAF intersect cirumcircle of ABCABC at points MM and NN. On rays ABAB and ACAC we have points PP and RR, such that angle PEAPEA is equal to angle BB and angle AERAER is equal to angle CC. Let LL be intersection of AEAE and PRPR and DD be intersection of BCBC and LNLN. Prove that 1∣MN∣+1∣EF∣=1∣ED∣.\frac{1}{|MN|}+\frac{1}{|EF|}=\frac{1}{|ED|}.
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