Geometry #3 (Analysis of its diagram and expansion)
by XmL, Jun 29, 2014, 3:34 AM
First of all it feels good to return to doing geometry again 
Problem:Two circle
,
meet at A,B.Denote MN is common tangent near 
of 
(
,
.A line parallel 
respectively meets , at 
. 
is intersection of 
.Prove that 
bisects angle 
I found this problem here: http://www.artofproblemsolving.com/Forum/viewtopic.php?f=46&t=594183
I will present some facts about this diagram that we can deduce from this set up. Note that they are all based on the assumption that are the outermost intersections with the circle.
Fun Fact 1:
.
We could've approached the original problem by proving this. However, this can be easily deduced from our set up, which contains a lot of cyclic quads and similar triangles.
Indeed, through some angle chasing we have
(side note: 
are actually corresponding points wrt the two triangles). and 
. Hence 
Fun Fact 2: The locus of is actually the circle of similitude of , aka the Appolonius circle of 
. Hence it immediately follows from fact 1.
Intrestingly this fact in turn gives rise to a well known property regarding circle of similitude of two circles: If lies on this circle, then 
.
Fun Fact 3: Unrelated to the previous two facts, It follows from radical axis that
are collinear.
Problem:Two circle
,
meet at A,B.Denote MN is common tangent near 
of 
(
,
.A line parallel 
respectively meets , at 
. 
is intersection of 
.Prove that 
bisects angle 
I found this problem here: http://www.artofproblemsolving.com/Forum/viewtopic.php?f=46&t=594183
.
are concyclic. Similarly 
are concyclic. Moreover, since 
,
are concyclic. Hence 
which is what we desire to prove.
denote the distance from 
to 
where 
is the radius of 
and then use Pythagorean theorem on 
:
.
:
same distance since 
.
.
are not collinear. Then by law of sines 
which is true since 
,I will present some facts about this diagram that we can deduce from this set up. Note that they are all based on the assumption that
are the outermost intersections with the circle.Fun Fact 1:
.We could've approached the original problem by proving this. However, this can be easily deduced from our set up, which contains a lot of cyclic quads and similar triangles.
Indeed, through some angle chasing we have
(side note: 
are actually corresponding points wrt the two triangles). and 
. Hence 
Fun Fact 2: The locus of
is actually the circle of similitude of , aka the Appolonius circle of 
. Hence it immediately follows from fact 1.Intrestingly this fact in turn gives rise to a well known property regarding circle of similitude of two circles: If
lies on this circle, then 
.Fun Fact 3: Unrelated to the previous two facts, It follows from radical axis that
are collinear.This post has been edited 2 times. Last edited by XmL, Aug 12, 2014, 11:16 PM
May 2016
April 2016