Journal of Geometry

, Volume 66, Issue 1, pp 72–103

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

• Henry Martyn Cundy
• Cyril Frederick Parry
Article

DOI: 10.1007/BF01225673

Cundy, H.M. & Parry, C.F. J Geom (1999) 66: 72. doi:10.1007/BF01225673

## 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.

### 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