Coaxal circles in incenter/excenter configuration
by liberator, Apr 15, 2015, 7:18 PM
Problem: Let 
be a triangle, whose excircle (opposite 
) touches 
at 
respectively. Denote 
as the intersection of the lines 
and 
, and 
as the intersection of the lines 
and 
. If 
is the excenter of 
, opposite , prove that the circumcircles of triangles 
and 
are coaxal.
Commentary: We may replace "excircle" with "incircle", "excenter" with "incenter", and the result still holds.
See interactive diagram here.
be a triangle, whose excircle (opposite 
) touches 
at 
respectively. Denote 
as the intersection of the lines 
and 
, and 
as the intersection of the lines 
and 
. If 
is the excenter of 
, opposite , prove that the circumcircles of triangles 
and 
are coaxal.Commentary: We may replace "excircle" with "incircle", "excenter" with "incenter", and the result still holds.
See interactive diagram here.
denote the second intersection of circles 
and 
.
.
is cyclic, from perpendicularity of tangents.
is tangent to 
and 
.
.
and 
,
.
and 
,
be the radical center of the three circles 
.
at 
,
.
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October 2018
April 2018