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Problem 4

Let ABCDABCD be a cyclic quadrilateral. Let P,Q,RP,Q,R be the feet of perpendiculars from DD to lines BC,CA,ABBC,CA,AB, respectively. Show that PQ=QRPQ=QR if and only if the bisectors of angles ABCABC and ADCADC meet on segment ACAC.
Step 2 of 7: Encode the midpoint by a cross-ratio
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In plain words

A midpoint is the harmonic relation with the point at infinity of its line.

cr⁡(P,R;Q,I∞)=−1\operatorname{cr}(P,R;Q,I_\infty)=-1
Detailed analysis

Let I∞I_\infty be the point at infinity in the direction of the Simson line, and use the affine convention cr⁡(X,Y;Z,I∞)=(z−x)/(z−y)\operatorname{cr}(X,Y;Z,I_\infty)=(z-x)/(z-y). Taking q=(p+r)/2q=(p+r)/2 gives cr⁡(P,R;Q,I∞)=−1\operatorname{cr}(P,R;Q,I_\infty)=-1, and conversely this equation gives 2q=p+r2q=p+r. Hence PQ=QR  ⟺  cr⁡(P,R;Q,I∞)=−1PQ=QR\iff\operatorname{cr}(P,R;Q,I_\infty)=-1.