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A, M, N, P are concyclic iff d passes through I- [VMO 2011]

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January 12, 2011
geometryincenterangle bisectorgeometry proposed

Problem Statement

Let △ABC\triangle ABC be a triangle such that ∠C\angle C and ∠B\angle B are acute. Let DD be a variable point on BCBC such that D≠B,CD\neq B, C and ADAD is not perpendicular to BC.BC. Let dd be the line passing through DD and perpendicular to BC.BC. Assume d∩AB=E,d∩AC=F.d \cap AB= E, d \cap AC =F. If M,N,PM, N, P are the incentres of △AEF,△BDE,△CDF.\triangle AEF, \triangle BDE,\triangle CDF. Prove that A,M,N,PA, M, N, P are concyclic if and only if dd passes through the incentre of △ABC.\triangle ABC.
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