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Problems
Contests
National and Regional Contests
Russia Contests
Sharygin Geometry Olympiad
2024 Sharygin Geometry Olympiad
4
4
Part of
2024 Sharygin Geometry Olympiad
Problems
(1)
Incircle Angle chase geo
Source: Sharygin Correspondence Round 2024 P4
3/6/2024
The incircle
ω
\omega
ω
of triangle
A
B
C
ABC
A
BC
touches
B
C
,
C
A
,
A
B
BC, CA, AB
BC
,
C
A
,
A
B
at points
A
1
,
B
1
A_1, B_1
A
1
,
B
1
and
C
1
C_1
C
1
respectively,
P
P
P
is an arbitrary point on
ω
\omega
ω
. The line
A
P
AP
A
P
meets the circumcircle of triangle
A
B
1
C
1
AB_1C_1
A
B
1
C
1
for the second time at point
A
2
A_2
A
2
. Points
B
2
B_2
B
2
and
C
2
C_2
C
2
are defined similarly. Prove that the circumcircle of triangle
A
2
B
2
C
2
A_2B_2C_2
A
2
B
2
C
2
touches
ω
\omega
ω
.
geometry