Generalization of Lester's Theorem
by TelvCohl, May 2, 2023, 9:26 AM
In this short post, I present a proof to a generalization of Lester's theorem found by Dao Thanh Oai [1].
Preliminaries
First, recall the following well-known
Lemma 1 : Given a
and two points 
Then 
and 
are conjugate WRT any conic passing through 
where is the cevian quotient of 
WRT 
In particular, 
is tangent to the circumconic of passing through and the circumconic of passing through 
Corollary 1 : Given a and two points 
Let 
be the crosspoint of 
WRT and let 
be the isogonal conjugate of 
WRT 
respectively. Then and 
are conjugate WRT any conic passing through 
Lemma 2 : Given a with orthocenter 
and a pair of isogonal conjugate 
of Let 
be the pedal triangle of 
WRT Then the crosspoint of 
WRT is the anticomplement of WRT 
Proof : Let
be the orthic triangle of 
be the anticomplement of WRT and let be the intersection of 
It suffices to prove that 
are collinear. Let 
be the reflection of in the midpoint of 
then 
and 
so note that 
we conclude that 
Lemma 3 : Given a rectangular hyperbola
with antipode Let 
be two points such that 
and are conjugate WRT and 
is parallel to the tangents of at Then 
are concyclic.
Proof : Let be the second intersection of 
with 
then by Seydewitz-Staudt theorem we get 
so note that is the isogonal conjugate of the perpendicular bisector of 
WRT 
we conclude that and are antiparallel WRT 
i.e. are concyclic. 
Main result
Property : Given a with Fermat point 
and a point lying on the Neuberg cubic of Let be the perspector of and the reflection triangle of WRT Then 
are concyclic.
Proof :
Let
be the circumcenter, symmedian point of respectively, be the isogonal conjugate of WRT and let 
be the intersection of 
Claim.
and are conjugate WRT the circumconic 
of passing through 
Proof. Let
be the centroid, orthocenter of respectively and let 
be the isogonal conjugate of WRT By Corollary 1, it suffices to show that the crosspoint of 
WRT lies on 
Since 
is the centroid of the pedal triangle of WRT respectively, so the line connecting and the centroid 
of the pedal triangle of WRT is parallel to 
hence from 
we get 
and the claim is proved by Lemma 2. 
Let
be the isogonal conjugate of WRT By New approach to Liang-Zelich Theorem (Sondat's theorem), lies on and the circum-rectangular hyperbola of passing through 
so is the second intersection of 
with and hence from Claim. we get the crosspoint of 
WRT lies on 
By Corollary 1, and are conjugate WRT the Kiepert hyperbola 
of Finally, the tangents of at are parallel to 
(see Character of the Steiner line WRT Cevian triangle (Corollary 1)), so by Lemma 3 we conclude that are concyclic.
Bibliography
[1]
Dao Thanh Oai, Three Conjectures in Euclidean Geometry, viXra:1507.0218, 2015.
Preliminaries
First, recall the following well-known
Lemma 1 : Given a
and two points 
Then 
and 
are conjugate WRT any conic passing through 
where is the cevian quotient of 
WRT 
In particular, 
is tangent to the circumconic of passing through and the circumconic of passing through 
Corollary 1 : Given a
and two points 
Let 
be the crosspoint of 
WRT and let 
be the isogonal conjugate of 
WRT 
respectively. Then and 
are conjugate WRT any conic passing through 
Lemma 2 : Given a
with orthocenter 
and a pair of isogonal conjugate 
of Let 
be the pedal triangle of 
WRT Then the crosspoint of 
WRT is the anticomplement of WRT 
Proof : Let
be the orthic triangle of 
be the anticomplement of WRT and let be the intersection of 
It suffices to prove that 
are collinear. Let 
be the reflection of in the midpoint of 
then 
and 
so note that 
we conclude that 
Lemma 3 : Given a rectangular hyperbola
with antipode Let 
be two points such that 
and are conjugate WRT and 
is parallel to the tangents of at Then 
are concyclic.Proof : Let
be the second intersection of 
with 
then by Seydewitz-Staudt theorem we get 
so note that is the isogonal conjugate of the perpendicular bisector of 
WRT 
we conclude that and are antiparallel WRT 
i.e. are concyclic. 
Main result
Property : Given a
with Fermat point 
and a point lying on the Neuberg cubic of Let be the perspector of and the reflection triangle of WRT Then 
are concyclic.Proof :
Let
be the circumcenter, symmedian point of respectively, be the isogonal conjugate of WRT and let 
be the intersection of 
Claim.
and are conjugate WRT the circumconic 
of passing through 
Proof. Let
be the centroid, orthocenter of respectively and let 
be the isogonal conjugate of WRT By Corollary 1, it suffices to show that the crosspoint of 
WRT lies on 
Since 
is the centroid of the pedal triangle of WRT respectively, so the line connecting and the centroid 
of the pedal triangle of WRT is parallel to 
hence from 
we get 
and the claim is proved by Lemma 2. 
Let
be the isogonal conjugate of WRT By New approach to Liang-Zelich Theorem (Sondat's theorem), lies on and the circum-rectangular hyperbola of passing through 
so is the second intersection of 
with and hence from Claim. we get the crosspoint of 
WRT lies on 
By Corollary 1, and are conjugate WRT the Kiepert hyperbola 
of Finally, the tangents of at are parallel to 
(see Character of the Steiner line WRT Cevian triangle (Corollary 1)), so by Lemma 3 we conclude that are concyclic. Bibliography
[1]
Dao Thanh Oai, Three Conjectures in Euclidean Geometry, viXra:1507.0218, 2015.
May 2023
May 2021