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Geometry of Rhombus and Four Incenters

Source: 2021 Taiwan TST Round 3 Mock Day 2 P3

May 4, 2021
geometrygeometry proposedrhombusincenter

Problem Statement

Let ABCD ABCD be a rhombus with center O. O. P P is a point lying on the side AB. AB. Let I, I, J, J, and L L be the incenters of triangles PCD, PCD, PAD, PAD, and PBC,PBC, respectively. Let H H and K K be orthocenters of triangles PLB PLB and PJA, PJA, respectively.
Prove that OI⊥HK. OI \perp HK.
Proposed by buratinogigle
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