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Greece Junior Math Olympiad
2015 Greece Junior Math Olympiad
4
4
Part of
2015 Greece Junior Math Olympiad
Problems
(1)
AB=AC iff CD=BC/2, perpendicular on tangent (Greece Junior 2015)
Source:
7/15/2019
Let
A
B
C
ABC
A
BC
be an acute triangle with
A
B
≤
A
C
AB\le AC
A
B
≤
A
C
and let
c
(
O
,
R
)
c(O,R)
c
(
O
,
R
)
be it's circumscribed circle (with center
O
O
O
and radius
R
R
R
). The perpendicular from vertex
A
A
A
on the tangent of the circle passing through point
C
C
C
, intersect it at point
D
D
D
. a) If the triangle
A
B
C
ABC
A
BC
is isosceles with
A
B
=
A
C
AB=AC
A
B
=
A
C
, prove that
C
D
=
B
C
/
2
CD=BC/2
C
D
=
BC
/2
. b) If
C
D
=
B
C
/
2
CD=BC/2
C
D
=
BC
/2
, prove that the triangle
A
B
C
ABC
A
BC
is isosceles.
geometry
isosceles
tangent