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National and Regional Contests
Bulgaria Contests
Bulgaria Team Selection Test
2020 Bulgaria Team Selection Test
6
6
Part of
2020 Bulgaria Team Selection Test
Problems
(1)
Excenters and concyclic
Source: Bulgarian TST 2020 P6
1/23/2021
In triangle
△
A
B
C
\triangle ABC
△
A
BC
,
B
C
>
A
C
BC>AC
BC
>
A
C
,
I
B
I_B
I
B
is the
B
B
B
-excenter, the line through
C
C
C
parallel to
A
B
AB
A
B
meets
B
I
B
BI_B
B
I
B
at
F
F
F
.
M
M
M
is the midpoint of
A
I
B
AI_B
A
I
B
and the
A
A
A
-excircle touches side
A
B
AB
A
B
at
D
D
D
. Point
E
E
E
satisfies
∠
B
A
C
=
∠
B
D
E
,
D
E
=
B
C
\angle BAC=\angle BDE, DE=BC
∠
B
A
C
=
∠
B
D
E
,
D
E
=
BC
, and lies on the same side as
C
C
C
of
A
B
AB
A
B
. Let
E
C
EC
EC
intersect
A
B
,
F
M
AB,FM
A
B
,
FM
at
P
,
Q
P,Q
P
,
Q
respectively. Prove that
P
,
A
,
M
,
Q
P,A,M,Q
P
,
A
,
M
,
Q
are concyclic.
geometry