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Problems
Contests
National and Regional Contests
Russia Contests
Sharygin Geometry Olympiad
2021 Sharygin Geometry Olympiad
14
14
Part of
2021 Sharygin Geometry Olympiad
Problems
(1)
Line XY touches the excircle opposite to A
Source: XVII Sharygin Correspondence Round, P14
3/2/2021
Let
γ
A
,
γ
B
,
γ
C
\gamma_A, \gamma_B, \gamma_C
γ
A
,
γ
B
,
γ
C
be excircles of triangle
A
B
C
ABC
A
BC
, touching the sides
B
C
BC
BC
,
C
A
CA
C
A
,
A
B
AB
A
B
respectively. Let
l
A
l_A
l
A
denote the common external tangent to
γ
B
\gamma_B
γ
B
and
γ
C
\gamma_C
γ
C
distinct from
B
C
BC
BC
. Define
l
B
,
l
C
l_B, l_C
l
B
,
l
C
similarly. The tangent from a point
P
P
P
of
l
A
l_A
l
A
to
γ
B
\gamma_B
γ
B
distinct from
l
A
l_A
l
A
meets
l
C
l_C
l
C
at point
X
X
X
. Similarly the tangent from
P
P
P
to
γ
C
\gamma_C
γ
C
meets
l
B
l_B
l
B
at
Y
Y
Y
. Prove that
X
Y
XY
X
Y
touches
γ
A
\gamma_A
γ
A
.
geometry
excircle