CGMO 2013 Problem 2

As shown in the figure below, $ABCD$ is a trapezoid, $AB \parallel CD$ . The sides $DA$ , $AB$ , $BC$ are tangent to $\odot O_1$ and $AB$ touches $\odot O_1$ at $P$ . The sides $BC$ , $CD$ , $DA$ are tangent to $\odot O_2$ , and $CD$ touches $\odot O_2$ at $Q$ . Prove that the lines $AC$ , $BD$ , $PQ$ meet at the same point.

size(200);
defaultpen(linewidth(0.8)+fontsize(10pt));
pair A=origin,B=(1,-7),C=(30,-15),D=(26,6);
pair bisA=bisectorpoint(B,A,D),bisB=bisectorpoint(A,B,C),bisC=bisectorpoint(B,C,D),bisD=bisectorpoint(C,D,A);
path bA=A--(bisA+100*(bisA-A)),bB=B--(bisB+100*(bisB-B)),bC=C--(bisC+100*(bisC-C)),bD=D--(bisD+100*(bisD-D));
pair O1=intersectionpoint(bA,bB),O2=intersectionpoint(bC,bD);
dot(O1^^O2,linewidth(2));
pair h1=foot(O1,A,B),h2=foot(O2,C,D);
real r1=abs(O1-h1),r2=abs(O2-h2);
draw(circle(O1,r1)^^circle(O2,r2));
draw(A--B--C--D--cycle);
draw(A--C^^B--D^^h1--h2);
label(" $A$ ",A,NW);
label(" $B$ ",B,SW);
label(" $C$ ",C,dir(350));
label(" $D$ ",D,dir(350));
label(" $P$ ",h1,dir(200));
label(" $Q$ ",h2,dir(350));
label(" $O_1$ ",O1,dir(150));
label(" $O_2$ ",O2,dir(300));

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