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Collinear Points and the Nine-Point Circle

Source: MMC 2014 Problem 4

June 8, 2014
geometrytrigonometryprojective geometrytrig identitiesLaw of Sinesgeometry proposed

Problem Statement

In triangle ABCABC let A′A', B′B', C′C' respectively be the midpoints of the sides BCBC, CACA, ABAB. Furthermore let LL, MM, NN be the projections of the orthocenter on the three sides BCBC, CACA, ABAB, and let kk denote the nine-point circle. The lines AA′AA', BB′BB', CC′CC' intersect kk in the points DD, EE, FF. The tangent lines on kk in DD, EE, FF intersect the lines MNMN, LNLN and LMLM in the points PP, QQ, RR. Prove that PP, QQ and RR are collinear.
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