MathDB
Problems
Contests
National and Regional Contests
Russia Contests
Sharygin Geometry Olympiad
2024 Sharygin Geometry Olympiad
16
16
Part of
2024 Sharygin Geometry Olympiad
Problems
(1)
Nonstandard incentre config
Source: Sharygin Correspondence Round 2024 P16
3/6/2024
Let
A
A
1
,
B
B
1
,
AA_1, BB_1,
A
A
1
,
B
B
1
,
and
C
C
1
CC_1
C
C
1
be the bisectors of a triangle
A
B
C
ABC
A
BC
. The segments
B
B
1
BB_1
B
B
1
and
A
1
C
1
A_1C_1
A
1
C
1
meet at point
D
D
D
. Let
E
E
E
be the projection of
D
D
D
to
A
C
AC
A
C
. Points
P
P
P
and
Q
Q
Q
on sides
A
B
AB
A
B
and
B
C
BC
BC
respectively are such that
E
P
=
P
D
,
E
Q
=
Q
D
EP = PD, EQ = QD
EP
=
P
D
,
EQ
=
Q
D
. Prove that
∠
P
D
B
1
=
∠
E
D
Q
\angle PDB_1 = \angle EDQ
∠
P
D
B
1
=
∠
E
D
Q
.
geometry
incenter