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Incentres in right triangle

Source: RMO 2015 Mumbai Region

September 3, 2018
geometry

Problem Statement

Let ABCABC be a right-angled triangle with ∠B=90∘\angle B = 90^\circ and let BDBD be the altitude from BB on to ACAC. Draw DE⊥ABDE \perp AB and DF⊥BCDF \perp BC. Let P,Q,RP, Q, R and SS be respectively the incentres of triangle DFC,DBF,DEBDF C, DBF, DEB and DAEDAE. Suppose S,R,QS, R, Q are collinear. Prove that P,Q,R,DP, Q, R, D lie on a circle.
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