MathDB
Problems
Contests
National and Regional Contests
Turkey Contests
Turkey Olympic Revenge
2024 Turkey Olympic Revenge
4
4
Part of
2024 Turkey Olympic Revenge
Problems
(1)
Geo with unnecessary condition
Source: Turkey Olympic Revenge 2024 P4
8/6/2024
Let the circumcircle of a triangle
A
B
C
ABC
A
BC
be
Γ
\Gamma
Γ
. The tangents to
Γ
\Gamma
Γ
at
B
,
C
B,C
B
,
C
meet at point
E
E
E
. For a point
F
F
F
on line
B
C
BC
BC
which is not on the segment
B
C
BC
BC
, let the midpoint of
E
F
EF
EF
be
G
G
G
. Lines
G
B
,
G
C
GB,GC
GB
,
GC
meet
Γ
\Gamma
Γ
again at points
I
,
H
I,H
I
,
H
respectively. Let
M
M
M
be the midpoint of
B
C
BC
BC
. Prove that the points
F
,
I
,
H
,
M
F,I,H,M
F
,
I
,
H
,
M
lie on a circle.Proposed by Mehmet Can Baştemir
radical axis
geometry