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sum cosA + cosB wanted, 2 angle bisectors and an altitude related

Source: Belarus TST 2009 1.3

June 13, 2020
geometryangle bisectoraltitudetrigonometry

Problem Statement

Points T,P,HT,P,H lie on the side BC,AC,ABBC,AC,AB respectively of triangle ABCABC, so that BPBP and ATAT are angle bisectors and CHCH is an altitude of ABCABC. Given that the midpoint of CHCH belongs to the segment PT,PT, find the value of cos⁡A+cos⁡B\cos A + \cos B
I. Voronovich
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