MathDB
Problems
Contests
International Contests
Balkan MO Shortlist
2019 Balkan MO Shortlist
G2
G2
Part of
2019 Balkan MO Shortlist
Problems
(1)
midpoint of segment in a triangle 75-60-45
Source: BMO SL 2019, G2
11/8/2020
Let be a triangle
△
A
B
C
\triangle ABC
△
A
BC
with
m
(
∠
A
B
C
)
=
7
5
∘
m(\angle ABC) = 75^{\circ}
m
(
∠
A
BC
)
=
7
5
∘
and
m
(
∠
A
C
B
)
=
4
5
∘
m(\angle ACB) = 45^{\circ}
m
(
∠
A
CB
)
=
4
5
∘
. The angle bisector of
∠
C
A
B
\angle CAB
∠
C
A
B
intersects
C
B
CB
CB
at point
D
D
D
. We consider the point
E
∈
(
A
B
)
E \in (AB)
E
∈
(
A
B
)
, such that
D
E
=
D
C
DE = DC
D
E
=
D
C
. Let
P
P
P
be the intersection of lines
A
D
AD
A
D
and
C
E
CE
CE
. Prove that
P
P
P
is the midpoint of segment
A
D
AD
A
D
.
geometry