Chase Angles, not Girls
by djmathman, Jul 23, 2015, 2:57 AM
IMO 2015.4 wrote:
Triangle 
has circumcircle 
and circumcenter 
. A circle 
with center 
intersects the segment 
at points 
and 
, such that 
, , , and 
are all different and lie on line in this order. Let 
and 
be the points of intersection of and , such that , , , , and lie on in this order. Let 
be the second point of intersection of the circumcircle of triangle 
and the segment 
. Let 
be the second point of intersection of the circumcircle of triangle 
and the segment 
.
Suppose that the lines
and 
are different and intersect at the point 
. Prove that lies on the line 
.
has circumcircle 
and circumcenter 
. A circle 
with center 
intersects the segment 
at points 
and 
, such that 
, , , and 
are all different and lie on line in this order. Let 
and 
be the points of intersection of and , such that , , , , and lie on in this order. Let 
be the second point of intersection of the circumcircle of triangle 
and the segment 
. Let 
be the second point of intersection of the circumcircle of triangle 
and the segment 
.Suppose that the lines
and 
are different and intersect at the point 
. Prove that lies on the line 
.Whoops this took me > 2 hours
In any case, it's not insanely difficult to draw, and few additional points are included anyway.
,
,
be the midpoint of 
.
(since 
)
.
we have 
.
Now let 
and 
.
and 
are the incenters of 
and 
respectively. As a result, 
Similarly, 
.
.
trivially. Note that 
to 
,
.
.
,
and 
as desired. 
This post has been edited 1 time. Last edited by djmathman, Jul 23, 2015, 3:34 AM
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