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2021-IMOC qualification
G1
G1
Part of
2021-IMOC qualification
Problems
(1)
AP^2+PX^2=BP^2+PY^2=CP^2+PZ^2, projections , circumcenter, incenter
Source: 2021 IMOC qualification problem, G1
12/30/2021
Let
O
O
O
be the circumcenter and
I
I
I
be the incenter of
△
\vartriangle
△
,
P
P
P
is the reflection from
I
I
I
through
O
O
O
, the foot of perpendicular from
P
P
P
to
B
C
,
C
A
,
A
B
BC,CA,AB
BC
,
C
A
,
A
B
is
X
,
Y
,
Z
X,Y,Z
X
,
Y
,
Z
, respectively. Prove that
A
P
2
+
P
X
2
=
B
P
2
+
P
Y
2
=
C
P
2
+
P
Z
2
AP^2+PX^2=BP^2+PY^2=CP^2+PZ^2
A
P
2
+
P
X
2
=
B
P
2
+
P
Y
2
=
C
P
2
+
P
Z
2
.
geometry
Circumcenter
incenter
projections
circumcircle