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Problems
Contests
National and Regional Contests
Mexico Contests
Mexico National Olympiad
1998 Mexico National Olympiad
5
5
Part of
1998 Mexico National Olympiad
Problems
(1)
PJ // AC iff BC^2 = AC· QC
Source: Mexican Mathematical Olympiad 1998 OMM P5
7/28/2018
The tangents at points
B
B
B
and
C
C
C
on a given circle meet at point
A
A
A
. Let
Q
Q
Q
be a point on segment
A
C
AC
A
C
and let
B
Q
BQ
BQ
meet the circle again at
P
P
P
. The line through
Q
Q
Q
parallel to
A
B
AB
A
B
intersects
B
C
BC
BC
at
J
J
J
. Prove that
P
J
PJ
P
J
is parallel to
A
C
AC
A
C
if and only if
B
C
2
=
A
C
ā
Q
C
BC^2 = AC\cdot QC
B
C
2
=
A
C
ā
QC
.
geometry
Tangents