## Conic passing through three points and tangent to two lines II

This is another case from that handled in Conic3Pts2Tangents.html . Here the conic is assumed to pass through points {A,B,C} and being tangent to to given lines {t,t'} at A, B respectively.
In fact, by the theorem discussed in DesarguesInvolution3.html , the involution f defined on an arbitrary line through C intersecting t' at X1, t at X2 and AB at X* and having f(X1) = X2 and f(X*) = X* (fixed), maps C to another (fourth) point X of the conic.
Similarly, taking a second line through C we find a fifth point Y on the conic and reduce the problem to the standard one of construction of a conic passing through five points: {A,B,C,D,E}.

Similarly can be handled the case of the conic passing through points {A,B,C} and being tangent to to given lines {t,t'}, t passing through A, but t' not passing through any of the given points.
In that case we draw line BC intersecting t, t' at X2, X1 respectively and determine the involution f on BC, interchanging X1, X2 and B, C. A fixed point X* of (f) is contained in the tangency chord of t and t'. A fourth point D is determined as intersection point of AX* and t'. The problem is reduced to the previous one.