Problem:
Let P be a given point inside quadrilateral ABCD. Points Q1​ and Q2​ are located within ABCD such that
∠Q1​BC=∠ABP,∠Q1​CB=∠DCP,∠Q2​AD=∠BAP,∠Q2​DA=∠CDP
Prove that Q1​Q2​​∥AB if and only if Q1​Q2​​∥CD.
Solution:
We will prove that the lines AB,CD, and Q1​Q2​​ are either concurrent or all parallel. Let X and Y denote the reflections of P across the lines AB and CD. We first claim that XQ1​=YQ1​ and XQ2​=YQ2​. Indeed, let Z be the reflection of Q1​ across BC. Then XB=PB,BQ1​=BZ, and
∠XBQ1​=∠XBA+∠ABQ1​=∠ABC=∠PBC+∠CBZ=∠PBZ
whence △XBQ1​≅△PBZ and thus XQ1​=PZ. Similarly YQ1​=PZ, and so XQ1​= YQ1​. In exactly the same way, we see that XQ2​=YQ2​, establishing the claim.
We conclude that the line Q1​Q2​​ is the perpendicular bisector of the segment XY. If AB∥CD, then XY⊥AB and it follows that Q1​Q2​​∥AB, as desired. If the lines AB and CD are not parallel, then let R denote their intersection. Since RX=RP=RY,R lies on the perpendicular bisector of XY and thus R,Q1​,Q2​ are collinear, as desired.
OR
This solution uses isogonal conjugates. Recall that two points S,T are isogonal conjugates with respect to △ABC if ∠SAB=∠CAT,∠SBC=∠ABT, and ∠SCA=∠BCT, with any two of these equalities implying the third.
If AB∥CD, then there is nothing to prove; thus we assume AB intersects CD in a point R. Then Q1​ and P are isogonal conjugates with respect to △RBC, whence ∠Q1​RB=∠CRP, and Q2​ and P are isogonal conjugates with respect to △RAD, whence ∠Q2​RA=∠DRP. Therefore ∠Q1​RB=∠Q2​RA=∠Q2​RB and the lines AB,CD,Q1​Q2​​ all intersect at R.
Remark: Although not needed for the problem as stated, here is an alternate proof that if AB∥CD, then Q1​Q2​​ is parallel to both. Extend BQ1​​ and BP to meet CD at points E and F, respectively. Then ∠BCP=∠Q1​CE and ∠PBC=∠ABQ1​=∠CEQ1​, and so △PBC∼△Q1​EC, whence PC/PB=Q1​C/Q1​E. Similarly △Q1​BC∼△PFC and PC/PF=Q1​C/Q1​B. We conclude that Q1​B/Q1​E=PF/PB. Similarly, extend AQ2​​ and AP to meet CD at G and H; then Q2​A/Q2​G=PH/PA=PF/PB=Q1​B/Q1​E, and it follows that Q1​Q2​​∥AB∥CD.
The problems on this page are the property of the MAA's American Mathematics Competitions