2000 IMO Problems/Problem 1
Problem
Two circles and
intersect at two points
and
. Let
be the line tangent to these circles at
and
, respectively, so that
lies closer to
than
. Let
be the line parallel to
and passing through the point
, with
on
and
on
. Lines
and
meet at
; lines
and
meet at
; lines
and
meet at
. Show that
.
Solution
Given a triangle,
and a point
in its interior, assume that the circumcircles of
and
are tangent to
. Prove that ray
bisects
.
Let the intersection of
and
be
. By power of a point,
and
, so
.
Let ray
intersect
at
. By our lemma,
,
bisects
. Since
and
are similar, and
and
are similar implies
bisects
.
Now, since
is parallel to
. But
is tangent to the circumcircle of
hence
and that implies
So
is isosceles and
.
By simple parallel line rules, =\angle{ABM}
\angle{BAM}=\angle{EAB}
\textit{ASA}
\triangle{ABM}
\triangle{ABE}$are congruent.
We know that$ (Error compiling LaTeX. Unknown error_msg)BE=BM=BDED
\triangle{EMD}
E
M
EM
PQ
MP = MQ
\triangle{EPQ}
EP = EQ$ .
See Also
2000 IMO (Problems) • Resources | ||
Preceded by First Question |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 2 |
All IMO Problems and Solutions |