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PQ passes through centroid, 3 centroids, <ACB =<CEP=<CFQ=90^o, CP=AP, CQ=BQ

Source: 2021 Dutch IMO TST 3.2

December 29, 2021
Centroidright triangleright anglesgeometry

Problem Statement

Let ABCABC be a right triangle with ∠C=90o\angle C = 90^o and let DD be the foot of the altitude from CC. Let EE be the centroid of triangle ACDACD and let FF be the centroid of triangle BCDBCD. The point PP satisfies ∠CEP=90o\angle CEP = 90^o and ∣CP∣=∣AP∣|CP| = |AP|, while point QQ satisfies ∠CFQ=90o\angle CFQ = 90^o and ∣CQ∣=∣BQ∣|CQ| = |BQ|. Prove that PQPQ passes through the centroid of triangle ABCABC.
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