In this short post, I present a new proof to a result of Ivan Zelich and Xuming Liang [1].
Preliminaries
Sondat's theorem : Given two perspective and orthologic triangles
and
such that the perpendicular from
to
resp. are concurrent at
and the perpendicular from
to
resp. are concurrent at
Then their perspector
lies on
and the circum-rectangular hyperbola
of
passing through
respectively.
Proof : Note that the second part is a particular case of a
well-known result, so it suffices to prove that
Let
be the orthocenter of
then
is parallel to the tangent of
at
so
Lemma : Given a
and a pair
of isogonal conjugate WRT
Let
be the incenter, A-excenter, B-excenter, C-excenter of
resp. and let
be a conic passing through
Then
lies on the polar of
WRT
Proof : Let
be the intersection of
with
respectively, then
so by Desargues involution theorem we know that the involution on
induced by the pencil of conics passing through
coincide with harmonic conjugate WRT
If
are the intersection of
with
then from the discussion above we get
and hence
lies on the polar of
WRT
Main result
Theorem : Given a with circumcenter 
and orthocenter 
Let 
be a pair of isogonal conjugate of such that 
and let 
be the intersection of 
For 
denote 
as the image of the pedal triangle of WRT under the homothety 
Then and are perspective if and only if 
where 
is undefined when 
Proof :
Claim. If 
then and are perspective for 
Proof. Let
be the pole of
WRT the circum-rectangular hyperbola
of the excentral triangle of
passing through
and let
be the intersection of
with
then it suffices to prove that
Let
be the intersection of
with
respectively. By
Lemma we know that
and
is the polar of
WRT
respectively, so
and hence
On the other hand, by Menelaus' theorem for
and
we get
and
so
If
are concurrent for some
then by
Sondat's theorem we know that their perspector lies on
and the circum-rectangular hyperbola of
passing through
so from
Claim. we conclude that
and
are perspective if and only if
Bibliography
[1]
Ivan Zelich and Xuming Liang, GENERALISATIONS OF THE PROPERTIES OF THE NEUBERG CUBIC TO THE EULER PENCIL OF ISOPIVOTAL CUBICS, INTERNATIONAL JOURNAL OF GEOMETRY Vol.
4 (2015), No. 2, 5 - 25.
This post has been edited 1 time. Last edited by TelvCohl, May 28, 2021, 1:30 AM
by mqoi_KOLA, Apr 12, 2025, 11:58 AM 
(ans- 
)
. Let 
and 
be the feet of the altitudes from 
to 
and from 
to 
respectively.
and 
as the reflections of 
and 
be the circumcircle of 
. Denote by 
the second intersection of line 
with 
the intersection of ray 
with 
is the circumcenter of 
can be inscribed within a circle.
, with 
, and the circle 
with center 
the midpoint of side 
intersects the circle 
and 
. Show that 
.
intersects 
. Points 
and 
are defined similarly. Line 
intersects 
at points 
, 
. The tangents at these points to 
, 
. Points 
, 
, 
, 
are defined similarly. Prove that points 
satisfying 
Prove that 
When does equality hold?
and 
such that the perpendicular from 
to 
resp. are concurrent at 
and the perpendicular from 
to 
resp. are concurrent at 
Then their perspector 
lies on 
and the circum-rectangular hyperbola 
of 
passing through 
respectively.
Let 
be the orthocenter of 
then 
is parallel to the tangent of 
at 
so 
and a pair 
of isogonal conjugate WRT 
Let 
be the incenter, A-excenter, B-excenter, C-excenter of 
resp. and let 
be a conic passing through 
Then 
lies on the polar of 
be the intersection of 
with 
respectively, then 
so by Desargues involution theorem we know that the involution on 
If 
are the intersection of 
then from the discussion above we get 
and hence 
and orthocenter 
Let 
be a pair of isogonal conjugate of 
and let 
be the intersection of 
For 
denote 
as the image of the pedal triangle of 
Then 
where 
is undefined when 
then 
be the pole of 
WRT the circum-rectangular hyperbola 
of the excentral triangle of 
be the intersection of 
with 
then it suffices to prove that 
Let 
be the intersection of 
respectively. By Lemma we know that 
and 
is the polar of 
WRT 
respectively, so 
and hence 
On the other hand, by Menelaus' theorem for 
and 
we get 
and 
so 
are concurrent for some 
then by Sondat's theorem we know that their perspector lies on 
so from Claim. we conclude that 
Ivan Zelich and Xuming Liang, GENERALISATIONS OF THE PROPERTIES OF THE NEUBERG CUBIC TO THE EULER PENCIL OF ISOPIVOTAL CUBICS, INTERNATIONAL JOURNAL OF GEOMETRY Vol. 4 (2015), No. 2, 5 - 25.
be the set of positive integers. Find all functions 
such that![\[ m^2 + f(n) \mid mf(m) +n \]](http://latex.artofproblemsolving.com/2/f/4/2f409d1de993f1af8fd839bb8e9f87a57e1b8608.png)
and 
.
May 2023
May 2021
where 
Then the locus of point 
respectively are concurrent is a circumconic of 
and let 
with 
If 
is the point such that 
then 
and 
are homothetic with center 
so 
and hence 
.![\[\frac{1}{a^2+155}+\frac{1}{b^2+155}+\frac{1}{c^2+155}\le \frac{11}{780}\]](http://latex.artofproblemsolving.com/4/2/7/427e12cbfdbbba6f26407dbb5c2430aee26c15eb.png)
When does the equality hold ?