by XmL, May 24, 2016, 5:06 AM
Generalization: Let

such that

are concyclic. Suppose

are two circles through

and

respectively, such that they are corresponding circles between similar triangles
. If
, prove that

are isogonals wrt
.
Note: Another way to characterize

is the property
Solution 1(Indirect):
Let

denote the inversion transformation
, then

under
.
Suppose
. Let

denote the isogonal conjugate of

wrt

and
. We can show through angle chasing that
. By more angle chasing we can obtain

or
. Since

(by inversion's preservation of concylic points), therefore
. Hence
, so

and

are isogonals.
Solution 2(Direct, at last):
Lemma: Two circles

intersect at two points
. Given a point

that satisfies

and doesn't lie on
. Prove that

are isogonals wrt
.
Proof: Let

denote the foot of the bisector of

on
. Thus the
-Apollonius circle of

(aka the circle of similitude of

passes through
. Since
, 
also bisects
. Hence the lemma is proven.
Main proof: Since
, therefore if
, then

lies on the circle of similitude of
. By the Lemma above, this means

are isogonals wrt
. Becuase

are isogonals wrt
, therefore

are isogonals wrt
. 
This post has been edited 1 time. Last edited by XmL, May 24, 2016, 5:07 AM
Reason: .