such that 
and 
with another positive integer 
,
is a perfect square. For example, if the number on the blackboard is 
,
,
,
,
.
,
with circumcenter 
,
and that line 
is not parallel to line 
be the point on the circumcircle of triangle 
is perpendicular to 
be the point on line 
such that 
.
and 
be the points on the circumcircle of triangle 
is a diameter, and 
and 
are parallel. Let 
be the midpoint of 
.
and 
are perpendicular.
and 
are perpendicular.
Y by GoJensenOrGoHome, Adventure10
Let 
be the intouch triangle of 
, which has incenter 
and circumcenter 
. Let 
be the circle tangent to 
at 
and tangent to 
at 
; define circles 
and points 
similarly. Prove that:
(a)
concur on 
. [EDIT: OMDS THIS IS THE ISOGONAL MITTENPUNKT! 
]
(b)
concur.
Both these points seem to be triangle centers. Do the results still hold after extraversion?
be the intouch triangle of 
, which has incenter 
and circumcenter 
. Let 
be the circle tangent to 
at 
and tangent to 
at 
; define circles 
and points 
similarly. Prove that:(a)
concur on 
. [EDIT: OMDS THIS IS THE ISOGONAL MITTENPUNKT! ](b)
concur.Both these points seem to be triangle centers. Do the results still hold after extraversion?
This post has been edited 1 time. Last edited by jlammy, Oct 23, 2015, 1:08 PM
Reason: omds X_57
Reason: omds X_57
(well-known), so if 
is the projection of 
it suffices to prove that 
are collinear and similarly for the others. Since 
bisects 
we have
