Inconics_In_Trilinears

1. Conics tangent to the triangle of reference

Here I follow closely Loney [LoneyII, p. 80] starting from the general equation of a conic in trilinear coordinates with respect
to a triangle of reference ABC (see Trilinears.html ). In this coordinate system the conic equation is of the form
(1) ax² + by² + cz² + 2hxy + 2fyz + 2gzx = 0.

The conic meets side BC at two points {A',A''} determined by solving equation (1) with x=0:
by² + cz² + 2fyz = 0.
The two solutions coincide exactly when the discriminant is zero i.e. f² = bc. Analogously the conic touches the other sides
when also g² = ca and h² = ab. Thus, writing {a,b,c} in the form of (possibly imaginary) squares {a=L², b=M², c=N²}
we obtain the form of the equation:

The ambiguities in sign are restricted by the requirement of non-degeneracy of the conic. Signs {+++, --+, -+-, +--} deliver
a double line i.e. an equation of the form (Ux+Vy+W)²=0, which also intersects the sides of the triangle of reference in two
coincident points. Thus only the sign combinations {---, -++, +-+, ++-} deliver genuine conics.

2. The group of four inconics

I call group of four inconics the set of the four conics obtained from one of them by changing the signs in the last equation.
To understand the difference I consider two possible cases:
(i)                                     L²x² + M²y² + N²z² - 2LMxy - 2MNyz -2NLzx = 0,
(ii)                                    L²x² + M²y² + N²z² + 2LMxy - 2MNyz +2NLzx = 0.
Intersecting with the line AC(y=0) we get the equations correspondingly:
                                         (Lx-Nz)² = 0    and    (Lx+Nz)² = 0,
determining the contact point on AC at x/z = N/L and x/z = -N/L correspondingly i.e. The two conics contact line AC at the
harmonic conjugate points with respect to {A,C} correspondingly:   (N, 0, L) and   (-N, 0, L).
Analogously the contact points of the two conics on line AB(z=0) are the conjugate points: (M, L, 0) and (-M, L, 0), whereas
the contact point for both on BC(x=0) is the same point (0, N, M).
This leads to the following interesting figure.

In this a point D with barycentric coordinates (L, M, N) not lying on the sides of the triangle defines its traces {A',B',C'}
on the sides of the triangle of reference. Points {A'', B'', C''} are the harmonic conjugates correspondingly of {A',B',C'} with
respect to {BC, CA, AB}. They have respectively coordinates {(-L, M, N), (L, -M, N), (L, M, -N)}. The four conics result
by replacing these values into the equation
(3) L²x² + M²y² + N²z² -2LMxy - 2MNyz - 2NLzx = 0.

In the figure are also drawn the two related conics (i) and (ii) and some additional relations are to be noticed:
(1) The passing of {AA', BB', CC'} through D,
(2) The location of the three points {A'', B'', C''} on a line.
Relation (1) results directly from Ceva (see Ceva.html and InconicsTangents.html ). Relation (2) defines the Trilinear polar
of D with respect to the triangle (see TrilinearPolar.html ).
Point D is called the perspector of conic (i) and its location completely determines the conic. There is an analogous
point D₁ for conic (ii) (its Perspector). It is again the intersection point of the three lines {AA', BB'', CC''}.
The other sign choices in the equation deliver two more conics analogous to (ii). The figure displays the corresponding
perspectors {D₁, D₂, D₃} each one of which completely determines the corresponding escribed conic.

The important fact is that inconics appear always in a group of fours. Anyone of the four suffices to determine the three others.
The conics are determined each by a single point: their perspector. The four corresponding perspectors are the vertices of
a (complete) quadrilateral {D,D₁,D₂,D₃} of which the triangle ABC is the diagonal one i.e. the triangle formed
by the intersections of its diagonals. One could also speak of the four conics defined by a quadrilateral.

3. The dual group of four circumconics

The dual conic can be defined through the inverse of the matrix of a conic. Since matrices differing by a (non-zero)
multiplicative factor determine the same conic, it even suffices to consider the adjoint matrix or any other (non-zero) multiple
of the inverse matrix. Here for the inconics the matrix is:

Note that the conic defined by this matrix has the form:
(4) Lyz + Mxz + Nxy = 0,
which is a conic passing through the vertices of the triangle of reference. Since every conic passing through the vertices
can be written in this form, the inverse correspondence of taking the inverse of the matrix m'' determines the dual conic of
the one defined by (4) which is the inconic defined by the matrix m.
Point D with coordinates (L,M,N) is also called the perspector of the conic (4).
Remark
A figure displaying the two dual conics, one inscribed and the other circumscribed to the triangle of reference, can be
viewed in TriangleConics.html . There it is seen that the circumconic (3) is also inscribed in the triangle of the harmonic
associates of the perspector D i.e. if {D₁, D₂, D₃} are the harmonic associates of D, then the circumconic (4)
is at the same time inconic (inscribed) in the triangle D₁D₂D₃. This allows the finding of the perspector D of
a circumconic through the tangential triangle of the conic i.e. the triangle formed by the tangents to the conic at the
vertices of the triangle of reference.

4. The norm equation

Traditionally the dual of the circum-conic (4) which is the in-conic satisfying equation (3) is written in the form

called the norm-equation [Casey, p. 121] of the inconic. This form underlines the duality between the two conics (3)
and (4) and the fact that they have both the same perspector, which is the point with barycentric coordinates (L,M,N).
Besides this equation leads to equation (3) by eliminating the radicals, for example sending the last term on the right side
and squaring, then isolating the square root and squaring again.
See the file IncircleInTrilinears.html for the computation of the equations of the incircle and excircles of the triangle of
reference.

Bibliography

[Casey] Casey, J. A Treatise on the Analytical Geometry of the Point, Line, Circle, and Conic Sections, 2nd ed. rev. enl. . Dublin, Hodges, Figgis, & Co., 1893
[LoneyII] Loney, S. L. Coordinate Geometry, vol. II Trilinear Coordinates Macmillan, London 1934

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1. Conics tangent to the triangle of reference

2. The group of four inconics

3. The dual group of four circumconics

4. The norm equation

See Also

Bibliography