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National and Regional Contests
Iran Contests
Iran MO (2nd Round)
2017 Iran MO (2nd Round)
6
6
Part of
2017 Iran MO (2nd Round)
Problems
(1)
An easy geometry from iran
Source: IRAN 2017 2nd round p6
4/21/2017
Let
A
B
C
ABC
A
BC
be a triangle and
X
X
X
be a point on its circumcircle.
Q
,
P
Q,P
Q
,
P
lie on a line
B
C
BC
BC
such that
X
Q
⊥
A
C
,
X
P
⊥
A
B
XQ\perp AC , XP\perp AB
XQ
⊥
A
C
,
XP
⊥
A
B
. Let
Y
Y
Y
be the circumcenter of
△
X
Q
P
\triangle XQP
△
XQP
. Prove that
A
B
C
ABC
A
BC
is equilateral triangle if and if only
Y
Y
Y
moves on a circle when
X
X
X
varies on the circumcircle of
A
B
C
ABC
A
BC
.
geometry
circumcircle