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exists triangle with sides D' P , E'Q and F'R , altitudes related

Source: Mathematics Regional Olympiad of Mexico Center Zone 2018 P6

November 13, 2021
triangle inequalitygeometryaltitudes

Problem Statement

Let △ABC\vartriangle ABC be a triangle with orthocenter HH and altitudes ADAD, BEBE and CFCF. Let D′D', E′E' and F′F' be the intersections of the heights ADAD, BEBE and CFCF, respectively, with the circumcircle of △ABC\vartriangle ABC , so that they are different points from the vertices of triangle △ABC\vartriangle ABC. Let LL, MM and NN be the midpoints of BCBC, ACAC and ABAB, respectively. Let P P, QQ and RR be the intersections of the circumcircle with LHLH, MHMH and NHNH, respectively, such that P P and A A are on opposite sides of BCBC, QQ and AA are on opposite sides of ACAC and RR and CC are on opposite sides of ABAB. Show that there exists a triangle whose sides have the lengths of the segments D′PD' P, E′QE'Q, and F′RF'R.
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