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perpendiculars given, parallel wanted

Source: Dutch NMO 2012 p4

September 6, 2019
geometryperpendicularparallel

Problem Statement

We are given an acute triangle ABCABC and points DD on BCBC and EE on ACAC such that ADAD is perpendicular to BCBC and BEBE is perpendicular to ACAC. The intersection of ADAD and BEBE is called HH. A line through HH intersects line segment BCBC in PP, and intersects line segment ACAC in QQ. Furthermore, KK is a point on BEBE such that PKPK is perpendicular to BEBE, and LL is a point on ADAD such that QLQL is perpendicular to ADAD. Prove that DKDK and ELEL are parallel. [asy] unitsize(1 cm);
pair A, B, C, D, E, H, K, L, P, Q;
A = (0,0); B = (6,0); C = (3.5,4); D = (A + reflect(B,C)*(A))/2; E = (B + reflect(A,C)*(B))/2; H = extension(A, D, B, E); P = extension(H, H + dir(-10), B, C); Q = extension(H, H + dir(-10), A, C); K = (P + reflect(B,E)*(P))/2; L = (Q + reflect(A,D)*(Q))/2;
draw(A--B--C--cycle); draw(A--D); draw(B--E); draw(K--P--Q--L); draw(rightanglemark(B,D,A,5)); draw(rightanglemark(B,E,A,5)); draw(rightanglemark(P,K,B,5)); draw(rightanglemark(A,L,Q,5));
dot("AA", A, SW); dot("BB", B, SE); dot("CC", C, N); dot("DD", D, NE); dot("EE", E, NW); dot("HH", H, N); dot("KK", K, SW); dot("LL", L, SE); dot("PP", P, NE); dot("QQ", Q, NW); [/asy]
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