Hagge-like circles, Jerabek hyperbola, Lemoine cubic
by kosmonauten3114, Jun 2, 2025, 4:05 PM
Let 
be a scalene oblique triangle with circumcenter 
and orthocenter 
, and 
(
, 
) a point in the plane.
Let
, 
be the circumcevian triangles of , , respectively.
Let
be the pedal triangle of with respect to .
Let
be the reflection in 
of 
. Define 
, 
cyclically.
Let
be the reflection in of 
. Define 
, 
cyclically.
Let
, 
be the circumcenters of 
, 
, respectively.
Prove that:
1), , are collinear if and only if lies on the Jerabek hyperbola of .
2), , are collinear if and only if lies on the Lemoine cubic (= 
) of .
be a scalene oblique triangle with circumcenter 
and orthocenter 
, and 
(
, 
) a point in the plane.Let
, 
be the circumcevian triangles of , , respectively.Let
be the pedal triangle of with respect to .Let
be the reflection in 
of 
. Define 
, 
cyclically.Let
be the reflection in of 
. Define 
, 
cyclically.Let
, 
be the circumcenters of 
, 
, respectively.Prove that:
1)
, , are collinear if and only if lies on the Jerabek hyperbola of .2)
, , are collinear if and only if lies on the Lemoine cubic (= 
) of .Attachments:
such that 
are one of such 
with its incircle 
.
and 
be the points of tangency of 
and 
respectively. On the edges 
and 
,
and 
is tangent to 
is the intersection point of the lines 
and 
.
and 
are parallel or coincide.
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be a triangle with incentre 
and circumcentre 
be the touchpoints of the incircle with 
,
,
is the Euler line of 
.
,
,
be fixed positive integers. There are 
ducks sitting in a
intersects the altitude from 
at point 
.
be the midpoints of 
,
intersects 
,
meets 
,
is cyclic.
and let 
be the midpoint of 
Suppose that 
different from 
such that 
Prove that the orthocenter of 
lies on the circumcircle of 
.
that satisfy
for all integers 
and rational numbers 
.
,
is 
.
April 2021
February 2021