# Geometrical properties of some Euler and circular cubics. Part 1

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## Abstract

*This sequel to our earlier paper (1995) continues the investigation of the Euler cubic curves therein defined, with particular reference to perspectivities and associated conics. Study of the circular cubic in this pencil, the Neuberg cubic, brings with it some discussion of the properties of circular cubics in general*.

## Keywords

Geometrical Property Early Paper## Glossary

## Named points

- Bennett
See Isoptic

- Brocard
Isogonal points P,Q in a triangle ABC for which all six angles PBC, PCA, PAB, QCB, QAC, QBA are equal

- De Longchamps
Image of orthocentre H in circumcentre O

- Euler
Point on circumcircle whose Simson line is parallel to OH

- Fermat
Points F for which ⦔BFC = ⦔CFA = ⦔ AFB (two such)

- Hessian
Points of intersection of the Apollonius circles; their pedal triangles are equilateral

- lsogonic
See Fermat

- Isodynamic
See Hessian

- Isoptic
Point at which the 4 circumcircles of the component triangles of a quadrangle subtend equal angles

- Lemoine
Isogonal conjugate (K) of centroid G

- Neuberg
Point on circumcircle whose Simson line is perpendicular to OH

- Steiner
Point on circumcircle whose Simson line is parallel to OK

- Symmedian
See Lemoine; common point of symmedian lines

## Point pairs

- Isogonal conjugates
Points P, ¯P such that their joins to each vertex of a triangle form an angle having the same bisectors as the angle of the triangle there. They are polar conjugate points with respect to all conies through the tritangent centres

- Isotomic conjugates
Points P, ¯P such that their joins to each vertex meet the opposite side at the ends of a segment with the same midpoint as the side. They are polar conjugates with respect to all conies through the centroid and the vertices of the anticomplementary triangle (see below)

## Named lines

- Brocard axis
OK, mediator of join of Brocard points, containing Hessian points

- Cevians of P
PA, PB, and PC

- Euler
OH, containing also centroid G, 9-point centre and De Longchamps point

- Simson line
Join of feet of perpendiculars from a point of the circumcircle to sides BC, CA, AB

- Symmedians
Reflexions of medians in the angle bisectors

## Named curves Conies

- Apollonius circles
Three circles, one through each vertex for which the other two vertices are inverse

- Jerabek's hyperbola
Rectangular, through ABCHO; isogonal conjugate of OH

- Kiepert's hyperbola
Rectangular, through ABCHG; isogonal conjugate of OK

- Nine-point circle
Through 3 diagonal points and 6 midpoints of the sides of quadrangle ABCH

- Steiner ellipse
Touching sides at midpoints

- Tritangent circles
Touching the three sides of a triangle, either externally or internally

## Cubics

- Darboux
Auto-isogonal with pivot De Longchamps point

- Euler
Member of pencil of auto-isogonal cubics with pivot on Euler line

- Feuerbach
Euler cubic with pivot at Nine-point centre

- Lucas
Locus of isotomic conjugate points whose join contains the isotomic conjugate of H

- McCay
Euler cubic with pivot at circumcentre O

- Neuberg
Locus of isogonal conjugates whose join is parallel to OH; Euler cubic with pivot at infinity on OH

- Ortho
Euler cubic with pivot at orthocentre H

- Thomson
Euler cubic with pivot at centroid G

## Triangles associated with a point P, given base triangle ABC

- Cevian
LMN, where AP meets BC at L, &c

- Anticevian
Triangle L′M′N′ for which ABC is the Cevian triangle of P; [AL, PL′] = −l, &c

- Complementary
A′B′C, where A′ is the midpoint of BC, &c. B′C is parallel to CB, &c

- Anticomplementary
A″B″C″, where B″AC″ is a line parallel to CB, &c

- Pedal
DEF, where PD is perpendicular to B C, &c

- Antipedal
Triangle D′E′F with respect to which ABC is the pedal triangle of P. E′AF is perpendicular to AP, &c

- Cyclopedal
XYZ, where AP meets circumcircle again at X,&c

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