Very nice theorem:
http://tube.geogebra.org/student/m542855
Let ABC be a triangle,

be the first (or secon) Fermat point, let

be the point on the Kiepert hyperbola. Let

be the point on line FK. The line through P and perpendicular to

meet

at
. Define

cyclically. Show that

is an equilateral triangle. This triangle homothety to the outer(or inner) Napoleon triangle.
Lemma 1: (USA TST 2006, Problem 6) Let

be a triangle. Triangles

and

are constructed outside of triangle

such that

and

and
. Segments

and

meet at
. Let

be the circumcenter of triangle
. Prove that
Telv Cohl's proof:
Let

be the circumcenter of

. Let

be the midpoint of
, respectively .
Easy to see

.
Since

are concyclic , so we get

. ...

Since

,
so we get

, hence

. ...
From

and

we get

, so from

and

.
Lemma 2:
Let

be a point out of

satisfy

.
Let

be a point out of

satisfy

.
Let

be a point out of

satisfy

. Then

.
Proof of the lemma 2:
Let

satisfy

and

.
Easy to see

.
From

, so combine with

we get

is the circumcenter of

, hence from lemma 1, we get

. i.e.
From the lemma 2 we get the following property about Kiepert triangle :
The pedal triangle of the isogonal conjugate of

WRT

and the Kiepert triangle with angle

are homothetic .
( Moreover, the homothety center of these two triangles is the Symmedian point of

! )
(1)
Let

be the orthocenter of
, respectively . (

also lie on the Kiepert hyperbola of

)
Easy to see all

are homothetic with center

, so it is suffices to prove the case when

coincide with

.
From Pascal theorem (for
) we get

.
Similarly, we can prove

and

,
so

and the pedal triangle of the isogonal conjugate of

WRT

are homothetic , hence from
(1) we get

and the outer (or inner) Napoleon triangle are homothetic .
Dual problem:
Let ABC be a triangle, let P be a point on the line

(or
). Three line through

and perpendicular to

meets the line

(or
) at
, define

cyclically. Show that A0B0C0 are an equilateral triangle homothety to Napoleon triangle and the homothetic center on the line

(or
)
Reference:
[1]
http://www.artofproblemsolving.com/Forum/viewtopic.php?f=48&t=622242
[2]
http://www.artofproblemsolving.com/Forum/viewtopic.php?f=48&t=621954
[3]
http://www.artofproblemsolving.com/Forum/viewtopic.php?f=46&t=148830
This post has been edited 13 times. Last edited by daothanhoai, Jan 25, 2015, 4:49 AM