MathDB
Problems
Contests
International Contests
Baltic Way
2022 Baltic Way
14
14
Part of
2022 Baltic Way
Problems
(1)
Config geo with a weird circle
Source: Baltic Way 2022/14
11/12/2022
Let
Γ
\Gamma
Γ
denote the circumcircle and
O
O
O
the circumcentre of the acute-angled triangle
A
B
C
ABC
A
BC
, and let
M
M
M
be the midpoint of the segment
B
C
BC
BC
. Let
T
T
T
be the second intersection point of
Γ
\Gamma
Γ
and the line
A
M
AM
A
M
, and
D
D
D
the second intersection point of
Γ
\Gamma
Γ
and the altitude from
A
A
A
. Let further
X
X
X
be the intersection point of the lines
D
T
DT
D
T
and
B
C
BC
BC
. Let
P
P
P
be the circumcentre of the triangle
X
D
M
XDM
X
D
M
. Prove that the circumcircle of the triangle
O
P
D
OPD
OP
D
passes through the midpoint of
X
D
XD
X
D
.
geometry