Home page | Catalogue | Classes | Tables | Glossary | Notations | Links | Bibliography | Thanks | Downloads | Related Curves

Let P be a point in the plane of the reference triangle ABC and denote by

cevP : the cevian triangle of P
acvP : the anticevian triangle of P
pedP : the pedal triangle of P
apdP : the antipedal triangle of P
ccvP : the circumcevian triangle of P
cacP : the circumanticevian triangle of P
refP : the reflection triangle of P in the sidelines of ABC
symP : the reflection triangle of P in the vertices of ABC

The following table gives the loci of P such that two of these triangles are orthologic. Note the frequent occurence of the McCay cubic K003 and see Table 22 for a generalization.

true means the property is true for any P. In this case, the orthology center(s) are given.

L denotes the line at infinity, C denotes the circumcircle. 3H is the union of three circum-hyperbolas centered on the cevian lines of X(25). Each one is tangent at two vertices of ABC to the symmedians and at the last vertex to the circumcircle. For any point on 3H, cacP is degenerate into a "flat" triangle.

 

ABC

cevP

acvP

pedP

apdP

ccvP

cacP

refP

cevP

K007

 

 

 

 

 

 

 

acvP

K002

a sextic Q027

 

 

 

 

 

 

pedP

true

P, P*

L, C and the Stammler quartic Q066

L, C and the quartic Q026

 

 

 

 

 

apdP

true

P, r(P,O)

true

P, see (3)

true

P, see (4)

L, C and K003

 

 

 

 

ccvP

L, C and K003

C and a septic

C and the septic Q025

L, C and K003

true

P, see (2)

 

 

 

cacP

3H and a sextic S1

3H and a nonic

3H and K002

L, 3H, a quartic Q2 and K102

L, C, 3H and a quintic

C, 3H, a quartic Q2 and a sextic

 

 

refP

true

P, P*

L, C and the Stammler quartic Q066

L, C and the quartic Q026

true

see (1)

L, C and K003

L, C and K003

L, 3H, a quartic Q2 and K102

 

symP

true

P, r(P,H)

L and K007

L and K002

true

P, r(P,P*)

true

P, r(P,O)

L, C and K003

L and the sextic

S1

true

P, r(P,P*)

Notes :

  • P* is the isogonal conjugate of P
  • r(P, X) is the reflection of P in X
  • (1) the centers of orthology are the orthocenters of the two triangles
  • (2) the triangles have only one center of orthology, they are perspective
  • (3) the other center of orthology Pcev is the complement in apdP of the reflection of P in O (Jean-Pierre Ehrmann)
  • (4) the other center of orthology Pacv is the anticomplement in apdP of the reflection of P in O (Jean-Pierre Ehrmann)
  • the Stammler quartic Q066 is the isogonal transform of the Stammler hyperbola.
  • the quartic Q2 is the isogonal transform of the polar circle.

 

Loci related to two orthologic cevian triangles

Let P be a fixed point not lying on the sidelines of the reference triangle ABC with cevian triangle PaPbPc. Recall that PaPbPc is the pedal triangle of some point Q if and only if P lies on the Lucas cubic K007 in which case Q lies on the Darboux cubic K004. Note that the line PQ always contains the de Longchamps point X(20).

Let M be a variable point with cevian triangle MaMbMc.

The locus of M such that the triangles PaPbPc and MaMbMc are orthologic is a circumcubic K(M).

For any point M on K(M), the perpendiculars dropped from the vertices of PaPbPc on the sidelines of MaMbMc concur at O1 and the perpendiculars dropped from the vertices of MaMbMc on the sidelines of PaPbPc concur at O2. These points O1, O2 are called the centers of orthology. The loci of O1, O2 when M traverses K(M) are two other cubics K(O1), K(O2).

Properties of K(M)

K(M) is a circumcubic passing through :

• the vertices Ga, Gb, Gc of the antimedial triangle where the tangents are concurrent

• P

• H÷ctP and ta(H÷P)

• U, V, W on the sidelines of ABC where U also lies on the perpendicular at A0 (see below) to PbPc

 

Properties of K(O1)

K(O1) contains :

• Pa, Pb, Pc

• the orthocenters H of ABC and Hp of PaPbPc

• A1, B1, C1 where A1 is the intersection of the perpendiculars at Pb, Pc to AB, AC respectively

• the infinite points of the altitudes of ABC (the three real asymptotes of K(O2) pass through the midpoints of PaPbPc)

 

Properties of K(O2)

K(O2) is a circumcubic passing through :

• the orthocenters H of ABC and Hp of PaPbPc

• A0, B0, C0 where A0 is the intersection of the perpendiculars at B, C to PaPb, PaPc respectively

• the infinite points of the altitudes of PaPbPc (the three real asymptotes of K(O1) pass through the midpoints of ABC)

***

An interesting special case

It is known that the cevian triangle PaPbPc of P is the pedal triangle of some point Q if and only if P lies on the Lucas cubic K007 in which case Q lies on the Darboux cubic K004. The three cubics above have the following additional properties :

K(M) is pK(G, tgQ) and contains G, tP, gQ, tgQ, agQ. Note that the pivot tgQ lies on K183.

K(O1) is a central cubic passing through Q, gQ with center the complement of Q with respect to the cevian triangle of P. K(O1) is a pK with respect to this latter triangle and its pivot is the crosspoint of H and gQ which is also the reflection of Q about its center.

K(O2) is pK(G/cgQ, agQ), a central cubic passing through gQ, cgQ (its center), agQ.

The following table gives a selection of such cubics.

P

Q

K(M)

K(O1)

K(O2)

X(2)

X(3)

K007

X(3), X(4), X(5), X(10), X(946), X(2883)

K004

X(4)

X(4)

K045

X(3), X(4), X(52), X(185), X(389)

K044

X(7)

X(1)

K034

X(1), X(4), X(65), X(942), X(950)

K033

X(8)

X(40)

K154

X(4), X(40), X(72), X(84)

X(4), X(84), X(1490)

X(20)

X(1498)

X(2), X(20), X(253), X(280), X(347)

K004

X(4), X(3183)

X(69)

X(20)

K235

X(4), X(20), X(64), X(1885)

X(4), X(64), X(185), X(1498), X(2883)

X(189)

X(84)

K133

X(4), X(40), X(84)

X(1), X(4), X(40), X(946), X(962)

X(253)

X(64)

X(2), X(20), X(253), X(3146)

X(4), X(20), X(64), X(3146)

X(4), X(20), X(3146)

X(329)

X(1490)

X(2), X(189), X(329)

X(4), X(1490)

X(4), X(3182)

X(1032)

X(3346)

X(2), X(1032), X(1498)

X(4), X(1498)

X(4), X(64), X(1498)

X(1034)

X(3345)

X(2), X(271), X(342), X(1034), X(1490)

X(4), X(1490)

X(4), X(84), X(1490)

Notes :

  • the center of K(O1) lies on a central cubic with center X(5) passing through X(3), X(4), X(5), X(389), X(942) whose asymptotes are the parallels at X(5) to the cevian lines of X(3).
  • the center of K(O2) lies on the complement of K004 and its pivot on the anticomplement of K004.