TriangleConics

1. Triangle conics

By this term I mean conics circumscribing a triangle ABC (i.e. passing through its vertices) or inscribed in a triangle (i.e. tangent to its side-lines). Such conics go always in pairs: one inscribed (c₁ in the figure) paired with one circumscribed (c₀ in the figure) and the relation between the two conics is very natural and based on the idea of trilinear polar tr(P) (see TrilinearPolar.html ) of a point P with respect to ABC. Point P and the trilinear polar L_P=tr(P) of P are called respectively perspector and perspectrix of the conic (both c₀ and c₁). For definitions and first elementary properties of the ideas used see the file Trilinears.html as well as IsogonalGeneralized.html .

To study the properties of the two conics a very convenient coordinate system is the one defined by the projective base {A,B,C,P}, in which these points have respectively the coordinates (1,0,0), (0,1,0), (0,0,1) and (1,1,1) (see ProjectiveBase.html ).

[1] The traces of P, points {A'',B'',C''} have respectively the representations {A''=B+C, B''=C+A, C''=A+B}.
[2] The harmonic associates of P, points {A',B',C'} have respectively the representations {A'=B+C-A, B'=C+A-B, C'=A+B}.
[3] The conjugates of A'',B'',C'' with respect to the vertices of the triangle, which are on the tripolar tr(P) of P have correspondingly the representation {A*=B-C, B*=C-A, C*=A-B-C}.
[4] The trilinear polar L_P=tr(P) of P has the equation x+y+z = 0.
[5] More general, the trilinear polar of any point Q(a,b,c)=aA+bB+cC has the equation (1/a)x+(1/b)y+(1/c)z = 0.
[6] Conic c₀ has the equation yz+zx+xy = 0.
[7] The equation of the conic c₁ is easily calculated considering it as a member of the bitangent family generated by lines AB (represented by z=0), AC (represented by y=0) and C''B'' (represented by x-y-z=0).
The conic has then the form yz - k(x-y-z)² = 0, and using that A'' is on this conic, k is seen to be 1/4.
Finally conic c₁ obtains the form -x²-y²-z²+2xy+2yz+2zx=0.
[8] The matrices C₁ of c₀ and C₂ of c₀ are adjoint to each other (each a multiple of the other's inverse):

This shows that each is the dual of the other i.e. it can be identified with the conic representing the tangents of the other (in the respective projective space).
[9] The family generated by the two conics: {k₁c₀ + k₂c₁} contains the member 4c₀ - c₁ = (x+y+z)² = 0, which is the trilinear polar L_p of P (considered as a degenerate double-line-conic).

2. General properties

[1] Conic c₀ is the locus of tripoles of lines through P.
In fact, such a line ax+by+cz=0 has a+b+c=0 (since P(1,1,1) is on it), hence its tripole S(1/a,1/b,1/c) satisfies equation [1.6].
[2] The tangents of c₁ are tripolars tr(Q) of points Q on the tripolar tr(P).
The calculation is contained in section 8 of IsogonalGeneralized.html .
[3] The tangents of c₀ are tripolars tr'(Q) (tr' denoting tripolars w.r. to A'B'C') of points Q on the tripolar tr(P).
Apply [2] since c₀ has to A'B'C' the relation c₁ has to ABC.
[4] For every point Q on tr(P) the two tripolars tr(Q) and tr'(Q) intersect on tr(P). Line tr(P) is the polar of P with respect to both conics c₀ and c₁.
This is discussed in IsotomicGeneral.html .
Introduce the generalized isogonal transformation which in the coordinates of section 1 has the form:
F(x,y,z) = (1/x,1/y,1/z).
[5] Circumconic c₀ is the image F(tr(P)) of the tripolar of P.
[6] Inconic c₁ is the image F''(tr(P)) under the corresponding isogonal transformation for triangle A''B''C''.
From the two last assertions the first is obvious and the second reduces to the first.
[7] The two conics c₀, c₁ have no points in common, their matrices are inverse (up to a factor) to each other and generate a family of conics in which tr(P) is contained as a double line.
This follows from the calculations of [1.8].
A quick proof of these properties can be also given by noticing that the two conics are images of the circum/in-circle of an equilateral under an appropriate homography F. Homography F is defined by the requirements (i) to map the vertices of the equilateral to the vertices of the triangle, (ii) to map the center of the equilateral to point P. The line at infinity is then mapped under F onto L_P=tr(P). This method is discussed in IsotomicGeneral.html .

1. Triangle conics

2. General properties

See Also