We are given an acute triangle ABC and points D on BC and E on AC such that AD is perpendicular to BC and BE is perpendicular to AC. The intersection of AD and BE is called H. A line through H intersects line segment BC in P, and intersects line segment AC in Q. Furthermore, K is a point on BE such that PK is perpendicular to BE, and L is a point on AD such that QL is perpendicular to AD. Prove that DK and EL 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("A", A, SW);
dot("B", B, SE);
dot("C", C, N);
dot("D", D, NE);
dot("E", E, NW);
dot("H", H, N);
dot("K", K, SW);
dot("L", L, SE);
dot("P", P, NE);
dot("Q", Q, NW);
[/asy] geometryperpendicularparallel