Let
![ABCD](/media/m/9/c/e/9ce25711ba18d9663b73c3580de4bf5a.png)
be a cyclic quadrilateral. Let
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
,
![Q](/media/m/4/5/c/45ce8d14aa1eb54f755fd8e332280abd.png)
,
![R](/media/m/4/d/7/4d76ce566584cfe8ff88e5f3e8b8e823.png)
be the feet of the perpendiculars from
![D](/media/m/7/0/0/7006c4b57335ab717f8f20960577a9ef.png)
to the lines
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
,
![CA](/media/m/a/a/e/aaec86bc003cfdb64d54116a4cabd387.png)
,
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
, respectively. Show that
![PQ=QR](/media/m/6/3/6/6369d1727102db735e1ae00bb56b1730.png)
if and only if the bisectors of
![\angle ABC](/media/m/c/9/2/c92dca0f4ca20d0ca087b59e09a26fa8.png)
and
![\angle ADC](/media/m/f/b/f/fbfb6106441edaacc077ce9fbb9f8b33.png)
are concurrent with
![AC](/media/m/6/4/7/647ef3a5d68f07d59d84afe03a9dc655.png)
.
%V0
Let $ABCD$ be a cyclic quadrilateral. Let $P$, $Q$, $R$ be the feet of the perpendiculars from $D$ to the lines $BC$, $CA$, $AB$, respectively. Show that $PQ=QR$ if and only if the bisectors of $\angle ABC$ and $\angle ADC$ are concurrent with $AC$.