MathDB
Problems
Contests
National and Regional Contests
Thailand Contests
Thailand Online MO
2021 Thailand Online MO
P4
P4
Part of
2021 Thailand Online MO
Problems
(1)
A-angle bisector and B-foot of altitude
Source: 2021 Thailand Online MO P4 (Mock TMO contest)
4/6/2021
Let
A
B
C
ABC
A
BC
be an acute triangle such that
∠
B
>
∠
C
\angle B > \angle C
∠
B
>
∠
C
. Let
D
D
D
and
E
E
E
be the points on the segments
B
C
BC
BC
and
C
A
CA
C
A
, respectively, such that
A
D
AD
A
D
bisects
∠
A
\angle A
∠
A
and
B
E
⊥
A
C
BE\perp AC
BE
⊥
A
C
. Finally, let
M
M
M
be the midpoint of the side
B
C
BC
BC
. Suppose that the circumcircle of
△
C
D
E
\triangle CDE
△
C
D
E
intersects
A
D
AD
A
D
again at a point
X
X
X
different from
D
D
D
. Prove that
∠
X
M
E
=
9
0
∘
−
∠
B
A
C
\angle XME = 90^{\circ} - \angle BAC
∠
XME
=
9
0
∘
−
∠
B
A
C
.
geometry