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Problems
Contests
National and Regional Contests
Korea Contests
Korea National Olympiad
2022 Korea National Olympiad
5
5
Part of
2022 Korea National Olympiad
Problems
(1)
Tangent Length Inside the Incircle
Source: KMO 2022 P5
10/29/2022
For a scalene triangle
A
B
C
ABC
A
BC
with an incenter
I
I
I
, let its incircle meets the sides
B
C
,
C
A
,
A
B
BC, CA, AB
BC
,
C
A
,
A
B
at
D
,
E
,
F
D, E, F
D
,
E
,
F
, respectively. Denote by
P
P
P
the intersection of the lines
A
I
AI
A
I
and
D
F
DF
D
F
, and
Q
Q
Q
the intersection of the lines
B
I
BI
B
I
and
E
F
EF
EF
. Prove that
P
Q
‾
=
C
D
‾
\overline{PQ}=\overline{CD}
PQ
=
C
D
.
geometry
incenter