[alogo] All conics circumscribing a triangle

Consider a triangle ABC inscribed in a conic. Then the intersection points of pairs of lines consisting by a side (f.e. BC) and the tangent to the opposite vertex (f.e. A) intersect at three points A*, B*, C* which are on the same line e. e coincides with the trilinear polar of point E with respect to triangle ABC, E being the pole of e w.r. to the conic (see PascalOnTriangles.html ). This implies that all conics circumscribing the fixed triangle ABC are parameterized by the location of the corresponding point E. Further the conic can be constructed immediately using a projectivity.

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To understand the argument consider first a circumscribed conic and the corresponding point E and line e, as described by the above reference. Then fix an equilateral triangle A'B'C' and its center E'. There is a unique projectivity H mapping A, B, C, E respectively to A', B', C', E'. Considering the harmonic tetrad (BCA*A'') and its image under H one sees that line e maps under H onto the line at infinity and A'' onto the middle of B'C' and the conic itself maps onto the circumcircle of A'B'C'. It follows that every conic circumscribing ABC can be obtained through the inverse projectivity F = H-1 and the image F(c) of the circumcircle of A'B'C'.
A particular case represents the circumcircle of triangle ABC. The corresponding E being the symmedian point and e being the Lemoine axis of the triangle. Another case of interest is the Steiner circumellipse. To obtain this conic one identifies E with the centroid (intersection point of medians) of ABC.
By considering the composition G=F'*F-1 of two such projectivities one can avoid the use of the equilateral. This method to define the circumconics of a triangle is discussed in TriangleCircumconics.html .

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