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Cono Sur Olympiad
2008 Cono Sur Olympiad
5
5
Part of
2008 Cono Sur Olympiad
Problems
(1)
Isosceles triangle with inscribed semicircle
Source: Cono Sur 2008 #5
11/17/2015
Let
A
B
C
ABC
A
BC
be an isosceles triangle with base
A
B
AB
A
B
. A semicircle
Γ
\Gamma
Γ
is constructed with its center on the segment AB and which is tangent to the two legs,
A
C
AC
A
C
and
B
C
BC
BC
. Consider a line tangent to
Γ
\Gamma
Γ
which cuts the segments
A
C
AC
A
C
and
B
C
BC
BC
at
D
D
D
and
E
E
E
, respectively. The line perpendicular to
A
C
AC
A
C
at
D
D
D
and the line perpendicular to
B
C
BC
BC
at
E
E
E
intersect each other at
P
P
P
. Let
Q
Q
Q
be the foot of the perpendicular from
P
P
P
to
A
B
AB
A
B
. Show that
P
Q
C
P
=
1
2
A
B
A
C
\frac{PQ}{CP}=\frac{1}{2}\frac{AB}{AC}
CP
PQ
=
2
1
A
C
A
B
.
geometry
cono sur