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IMOC 2019 G3 (PQ bisects segment BC, orthocenter, circumcircle related)

Source: https://artofproblemsolving.com/community/c6h1958463p13536764

March 21, 2020
geometryorthocentercircumcirclebisects segment

Problem Statement

Given a scalene triangle △ABC\vartriangle ABC has orthocenter HH and circumcircle Ω\Omega. The tangent lines passing through A,B,CA,B,C are ℓa,ℓb,ℓc\ell_a,\ell_b,\ell_c. Suppose that the intersection of ℓb\ell_b and ℓc\ell_c is DD. The foots of HH on ℓa,AD\ell_a,AD are P,QP,Q respectively. Prove that PQPQ bisects segment BCBC. https://4.bp.blogspot.com/-iiQoxMG8bEs/XnYNK7R8S3I/AAAAAAAALeY/FYvSuF6vQQsofASnXJUgKZ1T9oNnd-02ACK4BGAYYCw/s400/imoc2019g3.png
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