Consider then an arbitrary conic c

[1] All trilinear polars tr(X) of points X lying on such a conic are concurrent to a point P

[2] The trilinear polar L

Introduce the projective base with base points {A,B,C} and coordinator D. In this system the base points have correspondingly coordinates {(1,0,0),(0,1,0),(0,0,1)} and D has coordinates (1,1,1).

In this system also the trilinear polar tr(D) has an equation of the form x+y+z=0 and the conics (c

c

c

An arbitrary conic c

a/x + b/y + c/z = 0 (1) .

D(1,1,1) assumed to be on the conic implies that

a+b+c=0 (2),

which means that P

On the other side the trilinear polar of a point X(u,v,w) on the conic c

x/u + y/v + z/w = 0.

But a/u + b/v + c/w = 0 implies that this line passes through P

To prove [2] notice that all points on tr(D) have the property that their trilinear polars are tangent to c

The intersection point of line P*+tD with c

@(bc+2a

This can be seen easily to satisfy the equation (1) of conic c

Note that t(P

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TriangleConics.html

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