Conic3Pts2Tangents2

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.

Conic passing through three points and tangent to two lines II

See Also