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Czech-Polish-Slovak Match
2009 Czech-Polish-Slovak Match
3
3
Part of
2009 Czech-Polish-Slovak Match
Problems
(1)
Point of concurrency independent of incircle
Source: Czech-Polish-Slovak Match, 2009
8/20/2011
Let
ω
\omega
ω
denote the excircle tangent to side
B
C
BC
BC
of triangle
A
B
C
ABC
A
BC
. A line
ℓ
\ell
ℓ
parallel to
B
C
BC
BC
meets sides
A
B
AB
A
B
and
A
C
AC
A
C
at points
D
D
D
and
E
E
E
, respectively. Let
ω
′
\omega'
ω
′
denote the incircle of triangle
A
D
E
ADE
A
D
E
. The tangent from
D
D
D
to
ω
\omega
ω
(different from line
A
B
AB
A
B
) and the tangent from
E
E
E
to
ω
\omega
ω
(different from line
A
C
AC
A
C
) meet at point
P
P
P
. The tangent from
B
B
B
to
ω
′
\omega'
ω
′
(different from line
A
B
AB
A
B
) and the tangent from
C
C
C
to
ω
′
\omega'
ω
′
(different from line
A
C
AC
A
C
) meet at point
Q
Q
Q
. Prove that, independent of the choice of
ℓ
\ell
ℓ
, there is a fixed point that line
P
Q
PQ
PQ
always passes through.
geometry