MathDB
Problems
Contests
National and Regional Contests
Myanmar Contests
Myanmar IMO Training
2023 Myanmar IMO Training
3
3
Part of
2023 Myanmar IMO Training
Problems
(1)
Weird circle passes through a fixed point
Source: Own
6/21/2023
Let
△
A
B
C
\triangle ABC
△
A
BC
be a triangle such that
A
B
=
A
C
AB = AC
A
B
=
A
C
, and let its circumcircle be
Γ
\Gamma
Γ
. Let
ω
\omega
ω
be a circle which is tangent to
A
B
AB
A
B
and
A
C
AC
A
C
at
B
B
B
and
C
C
C
. Point
P
P
P
belongs to
ω
\omega
ω
, and lines
P
B
PB
PB
and
P
C
PC
PC
intersect
Γ
\Gamma
Γ
again at
Q
Q
Q
and
R
R
R
.
X
X
X
and
Y
Y
Y
are points on lines
B
R
BR
BR
and
C
Q
CQ
CQ
such that
A
X
=
X
B
AX = XB
A
X
=
XB
and
A
Y
=
Y
C
AY = YC
A
Y
=
Y
C
. Show that as
P
P
P
varies on
ω
\omega
ω
, the circumcircle of
△
A
X
Y
\triangle AXY
△
A
X
Y
passes through a fixed point other than
A
A
A
.
geometry
Fixed point