Conic3Pts2Tangents

Conic passing through three points and tangent to two lines

This problem, in the case the given two lines t, t' do not contain any of the given three points {A,B,C}, admits a simple solution by applying a special case of the Desargues involution theorem (see DesarguesInvolution3.html ).
In fact, by this theorem, the involution defined on line AC by the pair of points (A,C) and (F,G), F, G being the intersections of AC with t' and t respectively, has a fixed point Y on the chord joining the tangent points of t and t'.
Analogously the involution interchanging (A,B) and (H,I) on line AB has a fixed point X on the same chord. Thus, the chord XY of tangency is known and by intersecting it with t and t' we obtain two additional points D, E and reduce the problem to the one of construction of a conic passing through five points: {A,B,C,D,E}.

References

Doerrie, Heinrich 100 Great Problems of Elementary Mathematics New York, Dover, 1965, p. 277.

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Conic passing through three points and tangent to two lines

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References