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Iran-Singapore-Taiwan Friendly Math Competition
2023 IRN-SGP-TWN Friendly Math Competition
5
5
Part of
2023 IRN-SGP-TWN Friendly Math Competition
Problems
(1)
Three lines forming an isosceles triangle
Source: IRN-SGP-TWN 2023 friendly contest p5
7/16/2023
I
,
Ω
I,\Omega
I
,
Ω
are the incenter and the circumcircle of triangle
A
B
C
ABC
A
BC
, respectively, and the tangents of
B
,
C
B,C
B
,
C
to
Ω
\Omega
Ω
intersect at
L
L
L
. Assume that
P
≠
C
P\neq C
P
=
C
is a point on
Ω
\Omega
Ω
such that
C
I
,
A
P
CI,AP
C
I
,
A
P
, and the circle with center
L
L
L
and radius
L
C
LC
L
C
are concurrent. Let the foot from
I
I
I
to
A
B
AB
A
B
be
F
F
F
, the midpoint of
B
C
BC
BC
be
M
M
M
,
X
X
X
is a point on
Ω
\Omega
Ω
s.t.
A
I
,
B
C
,
P
X
AI,BC,PX
A
I
,
BC
,
PX
are concurrent. Prove that the lines
A
I
,
A
X
,
M
F
AI,AX,MF
A
I
,
A
X
,
MF
form an isosceles triangle.Proposed by ckliao914
geometry