Difference between revisions of "2000 IMO Problems/Problem 1"
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Revision as of 00:04, 19 November 2023
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 it's 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
.
By simple parallel line rules, . Similarly,
, so by
criterion,
and
are congruent.
Now, and
since
is parallel to
. But
is tangent to the circumcircle of
hence
and that implies
So
is isosceles and
.
Join points and
,
is perpendicular on
(why?), previously we proved
, hence
is isoscles and
.
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 |