IMO Shortlist 1995 Geometry #3
by KingSmasher3, Apr 3, 2013, 5:29 AM
The incircle of triangle 
touches the sides 
, 
, 
at 
respectively. 
is a point inside triangle of such that the incircle of triangle 
touches at 
, and touches 
and 
at 
and 
respectively.
Show that
are concyclic.
________
Solution 1: Consider the circle centered at
passing through 
and 
and the circle centered at 
passing through 
and 
Their radical axis is the line through perpendicular to 
This line contains 
and 
the incenters of 
and 
respectively. Let 
intersect this line at the point 
Let 
and 
Using Menelaus on 
and 
see find that and 
are collinear if and only if 
which is clearly true since 
and 
This, in addition to the fact that is on the radical axis of the two circles centered at and 
shows that 
so by power of a point, 
is cyclic.
Solution 2: Extend
to meet 
at 
Using Menelaus of and 
we see that 
and 
are collinear. By power of a point, it follows that 
so is cyclic as desired.
Main Point(s): Menelaus can be used to prove collinearity (or concurrency) when applying power of a point.
touches the sides 
, 
, 
at 
respectively. 
is a point inside triangle of such that the incircle of triangle 
touches at 
, and touches 
and 
at 
and 
respectively.Show that
are concyclic.________
Solution 1: Consider the circle centered at
passing through 
and 
and the circle centered at 
passing through 
and 
Their radical axis is the line through perpendicular to 
This line contains 
and 
the incenters of 
and 
respectively. Let 
intersect this line at the point 
Let 
and 
Using Menelaus on 
and 
see find that and 
are collinear if and only if 
which is clearly true since 
and 
This, in addition to the fact that is on the radical axis of the two circles centered at and 
shows that 
so by power of a point, 
is cyclic.Solution 2: Extend
to meet 
at 
Using Menelaus of and 
we see that 
and 
are collinear. By power of a point, it follows that 
so is cyclic as desired.Main Point(s): Menelaus can be used to prove collinearity (or concurrency) when applying power of a point.
This post has been edited 2 times. Last edited by KingSmasher3, Apr 3, 2013, 7:32 PM
March 2014
October 2013