Problem 4

Let

ABCD

be a cyclic quadrilateral. Let

P,Q,R

be the feet of perpendiculars from

D

to lines

BC,CA,AB

, respectively. Show that

PQ=QR

if and only if the bisectors of angles

ABC

and

ADC

meet on segment

AC

Step 4 of 7: Project the cross-ratio to the circumcircle

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In plain words

Central projection preserves cross-ratios and turns the midpoint condition into a harmonic pencil at B.

\operatorname{cr}(P,R;Q,I_\infty)=\operatorname{cr}(BC,BA;BF,BE)

Detailed analysis

Project the four points $P,R,Q,I_\infty$ from $B$ to the pencil of lines through $B$ . Their images are $BC,BA,BF,BE$ , respectively: $BP=BC$ , $BR=BA$ , $BQ=BF$ , and Step 3 gives $BI_\infty=BE$ . Central projection is fractional-linear on an affine coordinate, so it preserves cross-ratio and gives $\operatorname{cr}(P,R;Q,I_\infty)=\operatorname{cr}(BC,BA;BF,BE)$ . Thus the metric midpoint condition has an exact projective formulation.