Journal of Geometry

, Volume 66, Issue 1–2, pp 72–103 | Cite as

Geometrical properties of some Euler and circular cubics. Part 1

  • Henry Martyn Cundy
  • Cyril Frederick Parry


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.


Geometrical Property Early Paper 
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Named points


See Isoptic


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


Point on circumcircle whose Simson line is parallel to OH


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


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


See Fermat


See Hessian


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


Isogonal conjugate (K) of centroid G


Point on circumcircle whose Simson line is perpendicular to OH


Point on circumcircle whose Simson line is parallel to OK


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


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


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



Auto-isogonal with pivot De Longchamps point


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


Euler cubic with pivot at Nine-point centre


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


Euler cubic with pivot at circumcentre O


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


Euler cubic with pivot at orthocentre H


Euler cubic with pivot at centroid G

Triangles associated with a point P, given base triangle ABC


LMN, where AP meets BC at L, &c


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


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


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


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


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


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


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Copyright information

© Birkhäuser Verlag 1999

Authors and Affiliations

  • Henry Martyn Cundy
    • 1
  • Cyril Frederick Parry
    • 2
  1. 1.KendalUK
  2. 2.ExmouthUK

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