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cyclic wanted, AF = FM = JD = DK = LE = EA, incircle related

Source: 2014 Saudi Arabia Pre-TST 4.4

September 12, 2020
geometryCyclicConcyclicincircleequal segments

Problem Statement

Let △ABC\vartriangle ABC be an acute triangle, with ∠A>∠B≥∠C\angle A> \angle B \ge \angle C. Let D,ED, E and FF be the tangency points between the incircle of triangle and sides BC,CA,ABBC, CA, AB, respectively. Let JJ be a point on (BD)(BD), KK a point on (DC)(DC), LL a point on (EC)(EC) and MM a point on (FB)(FB), such that AF=FM=JD=DK=LE=EA.AF = FM = JD = DK = LE = EA.Let PP be the intersection point between AJAJ and KMKM and let QQ be the intersection point between AKAK and JLJL. Prove that PJKQPJKQ is cyclic.
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