Gee? Woah... much intrigue
by shiningsunnyday, Jun 24, 2016, 2:50 PM
2000 IMO P1 wrote:
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 
.
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 
.
and all its congruent angles as 
and 
as 
Angle chasing as well as using the well-known tangent angle lemma yields 
Lemma 1: 
is cyclic.
Lemma 2: 
and 
Also, 
For the second pair, 
Also, 
Lemma 3: 
and 
For the second pair, apply the second part of Lemma 2: 
and 
To show 
it suffices to show 
Note that 
So it suffices to show that 
or that 
This can be shown as 
(since 
Hence KA=KB.
Hence 
Attachments:
This post has been edited 4 times. Last edited by shiningsunnyday, Jun 25, 2016, 5:38 AM
It follows that 
is the midpoint of 
so considering a homothety centered at 
we have that 
is the midpoint of 
and now it remains to show 
by parallel lines and Fact 5. Similarly, 
so 
as desired.
,
)
.
May 2017
April 2017