Let
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
be a triangle, and
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
a point in the interior of this triangle. Let
![D](/media/m/7/0/0/7006c4b57335ab717f8f20960577a9ef.png)
,
![E](/media/m/8/b/0/8b01e755d2253cb9a52f9e451d89ec11.png)
,
![F](/media/m/3/e/8/3e8bad5df716d332365fca76f53c1743.png)
be the feet of the perpendiculars from the point
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.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. Assume that
![AP^{2}+PD^{2}=BP^{2}+PE^{2}=CP^{2}+PF^{2}](/media/m/3/a/f/3af1ddf1e11a67453ecb842380971773.png)
.
Furthermore, let
![I_{a}](/media/m/c/8/7/c87798e71553983ab4acf237fee0be44.png)
,
![I_{b}](/media/m/9/7/7/977ac4038f40c7a7dc1968a1d19849f2.png)
,
![I_{c}](/media/m/1/d/8/1d8878d1a5d3e336f09e2f6f07fba90b.png)
be the excenters of triangle
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
. Show that the point
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
is the circumcenter of triangle
![I_{a}I_{b}I_{c}](/media/m/5/0/2/502f53b8042112499f1c954fa5e0da6e.png)
.
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Let $ABC$ be a triangle, and $P$ a point in the interior of this triangle. Let $D$, $E$, $F$ be the feet of the perpendiculars from the point $P$ to the lines $BC$, $CA$, $AB$, respectively. Assume that
$AP^{2}+PD^{2}=BP^{2}+PE^{2}=CP^{2}+PF^{2}$.
Furthermore, let $I_{a}$, $I_{b}$, $I_{c}$ be the excenters of triangle $ABC$. Show that the point $P$ is the circumcenter of triangle $I_{a}I_{b}I_{c}$.