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Find measure of angles if equilateral

Source: APMO 2002

April 8, 2006
geometry unsolvedgeometry

Problem Statement

Let ABCABC be an equilateral triangle. Let PP be a point on the side ACAC and QQ be a point on the side ABAB so that both triangles ABPABP and ACQACQ are acute. Let RR be the orthocentre of triangle ABPABP and SS be the orthocentre of triangle ACQACQ. Let TT be the point common to the segments BPBP and CQCQ. Find all possible values of ∠CBP\angle CBP and ∠BCQ\angle BCQ such that the triangle TRSTRS is equilateral.
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