Consider three lines, defining the triangle ABC. Let X, Y be points, considered fixed,

and P a movable point. Let A', B', C' the intersection-points of the lines AP, BP, CP, with the opposite sides BC, CA, AB, correspondingly. Let f be the conic passing through the three points A', B', C' and the two fixed X, Y. Let A'', B", C" the second intersection-points of the lines with the conic correspondingly. The lines AA'', BB'', CC'' meet all in one point P'. This defines the generalized Terquem map F : P --- > P'. When X, Y are the circular points at infinity the conic is a circle and the map coincides with the usual Terquem map. Find fixed points, dependence on X, Y etc.

Similarly one could consider instead of two points X, Y, a point and a tangent or two tangents, and let pass the conic through them. Particular points and lines. Images of circles, lines under this map.

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