New Results for Orthocenters
by XmL, Dec 22, 2015, 7:30 PM
Given
with isogonal conjugates
and orthocenter
.
Property1: Reflect
over
to get
, Prove that the lines through
perpendicular to
resp. are concurrent with
.
Proof: Denote the two lines defined in the statement
resp. Reflect
over
to get
which lies on the circumcircle of
, with center
. Let
. It is not hard to show that
are symmetric over
. If
, then it suffices to show
concur, which is equivalent to
are concyclic by the radical axis theorem
Let
be the isogonal conjugate of
wrt
. By angle relations of conjugates,
. Define
, thus
because
. Hence
is cyclic, and it suffices to show
.
Define
the reflection of
over
. Since
are isogonals wrt
, therefore
. Hence
, i.e
. Because
, it has been shown that
.
Property2: If
, then
lies on the radical axis of
. The second circle is also known as the circumcircle of the reflection triangle of
wrt
.
Proof: It suffices to show that the intersections of
with
and
with
are concyclic
the perpendicular bisectors of the two intersection segments is concurrent with the perpendicular bisector of
, i.e
. This is equivalent to Property1, thus Property2 is proven. 
Property3: Define
,
,
. Then
are collinear and
.
This is immediate from property2, since
lie on the radical axis of
. 
Theorem1:
, where
denote the circumradius of the reflection triangle of
wrt
.
Proof: Let the radical axis of
intersect
at
. Then
. If
, then it suffices to show that
.
Let the lines through
perpendicular to
resp. meet at
; from property1,
. Define
,
, then
by property3. Hence the multiple cyclic quadrilaterals give
. 



Property1: Reflect






Proof: Denote the two lines defined in the statement












Let









Define











Property2: If





Proof: It suffices to show that the intersections of








Property3: Define





This is immediate from property2, since



Theorem1:




Proof: Let the radical axis of




![$\implies R_{PQ}^2-HP^2-HQ^2=KQ^2-KH^2-HQ^2=QH[KQ+KH-HQ]=2QH\cdot HK$](http://latex.artofproblemsolving.com/f/b/5/fb5306edaa3708be795e72eaff6cb0f447479449.png)


Let the lines through









This post has been edited 1 time. Last edited by XmL, Jul 28, 2017, 7:08 AM
Reason: edited title
Reason: edited title