The following question consists of a final exam, which is originally written in Spanish, so please excuse me if something is lost in traslation. The problem goes as follows: We start of with a rhombus ABCD counterclockwise, where we define point P variable on the diagonal AC. We are then defined point Q as the intersection between the circumscribed circles of ABP and CDP. We are asked to prove that D,Q and B are colinear, in other words, that Q belongs to the diagonal CD.
Ive tried to tackle the problem through congruent triangles, or angles and paralel lines, but I cant seem to notice were to make the breakthrough to the solution. The only major conclusion until now is that triangle BPD is isosceles.
(1) Will $\angle 1 = \angle 2 =\angle 3$?
(2) Wll $\angle red = 90^0 - \angle 2 = 90^0 - \angle 3 = \angle purple$?
(3) Will the green marked angles equal?
If the answers are all yes, then you have all the necessary ingredients.