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Circumradius and heights

Source: 2014 China TST 2 Day 2 Q4

March 20, 2014
geometrycircumcirclegeometric transformationreflectiontrigonometrytrig identitiesLaw of Sines

Problem Statement

Given circle OO with radius RR, the inscribed triangle ABCABC is an acute scalene triangle, where ABAB is the largest side. AHA,BHB,CHCAH_A, BH_B,CH_C are heights on BC,CA,ABBC,CA,AB. Let DD be the symmetric point of HAH_A with respect to HBHCH_BH_C, EE be the symmetric point of HBH_B with respect to HAHCH_AH_C. PP is the intersection of AD,BEAD,BE, HH is the orthocentre of △ABC\triangle ABC. Prove: OP⋅OHOP\cdot OH is fixed, and find this value in terms of RR.
(Edited)
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