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National and Regional Contests
Bulgaria Contests
Bulgaria MO Regional Round
2024 Bulgaria MO Regional Round
12.1
12.1
Part of
2024 Bulgaria MO Regional Round
Problems
(1)
Conditional geo with incenters
Source: Bulgaria MO Regional round 2024, 12.1
2/13/2024
Let
A
B
C
ABC
A
BC
be an acute triangle with midpoint
M
M
M
of
A
B
AB
A
B
. The point
D
D
D
lies on the segment
M
B
MB
MB
and
I
1
,
I
2
I_1, I_2
I
1
,
I
2
denote the incenters of
△
A
D
C
\triangle ADC
△
A
D
C
and
△
B
D
C
\triangle BDC
△
B
D
C
. Given that
∠
I
1
M
I
2
=
9
0
∘
\angle I_1MI_2=90^{\circ}
∠
I
1
M
I
2
=
9
0
∘
, show that
C
A
=
C
B
CA=CB
C
A
=
CB
.
geometry