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
. By power of a point,
and
, so
.
Given a triangle, 
and a point 
and 
are tangent to 
. Prove that ray 
bisects 
Let the intersection of 
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=BD
ED
\triangle{EMD}
EM
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 | ||
