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 6 of 7: Apply the two angle-bisector theorems

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In plain words

Each internal bisector selects one point of AC by a side-length ratio.

\frac{AX}{XC}=\frac{AB}{BC},\qquad \frac{AY}{YC}=\frac{AD}{DC},\qquad X=Y\iff\frac{AD}{DC}=\frac{AB}{BC}

Detailed analysis

Let the internal bisector at $B$ meet $AC$ at $X$ , and the internal bisector at $D$ meet $AC$ at $Y$ . The angle-bisector theorem gives $\frac{AX}{XC}=\frac{AB}{BC}$ and $\frac{AY}{YC}=\frac{AD}{DC}$ . Both $X$ and $Y$ lie on the segment $AC$ and are uniquely determined by their positive ratios. Hence the bisectors meet at one point of $AC$ exactly when $X=Y\iff\frac{AD}{DC}=\frac{AB}{BC}$ .