equal segments inside a quadr. with 2 parallel sides, angle bisectors related
Source: Dutch NMO 2015 p3 juniors
September 7, 2019
geometryangle bisectorequal segmentsparallel
Problem Statement
In quadrilateral sides and are parallel. In each of the four vertices we draw an angular bisector. The angular bisectors of angles and intersect in point , those of angles and intersect in point , those of angles and intersect in point , and those of angles and intersect in point S. Suppose that is parallel to . Prove that .[asy]
unitsize(1.2 cm);pair A, B, C, D, P, Q, R, S;A = (0,0);
D = (3,0);
B = (0.8,1.5);
C = (3.2,1.5);
S = extension(A, incenter(A,B,D), D, incenter(A,C,D));
Q = extension(B, incenter(A,B,C), C, C + incenter(A,B,D) - A);
P = extension(A, S, B, Q);
R = extension(D, S, C, Q);draw(A--D--C--B--cycle);
draw(B--Q--C);
draw(A--S--D);dot("", A, SW);
dot("", B, NW);
dot("", C, NE);
dot("", D, SE);
dot("", P, dir(90));
dot("", Q, dir(270));
dot("", R, dir(90));
dot("", S, dir(90));
[/asy]Attention: the figure is not drawn to scale.