To study the properties of the two conics a very convenient coordinate system is the one defined by the

[1] The

[2] The

[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

[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 c

[7] The equation of the conic c

The conic has then the form yz - k(x-y-z)

Finally conic c

[8] The matrices C

This shows that each is the

[9] The family generated by the two conics: {k

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 c

The calculation is contained in section 8 of IsogonalGeneralized.html .

[3] The tangents of c

Apply [2] since c

[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 c

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 c

[6] Inconic c

From the two last assertions the first is obvious and the second reduces to the first.

[7] The two conics c

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

IncircleTangents2.html

InconicsTangents.html

IsogonalGeneralized.html

IsotomicGeneral.html

ProjectiveBase.html

PascalOnTriangles.html

ProjectiveCoordinates.html

ProjectivityFixingVertices.html

Steiner_Ellipse.html

TriangleCircumconics.html

TriangleCircumconics2.html

TriangleProjectivitiesPlay.html

TrilinearPolar.html

Trilinears.html

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