## 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}.

Conic3Pts2Tangents2.html
CrossRatio0.html
DesarquesInvolution.html
DesarquesInvolution2.html
DesarquesInvolution3.html
HomographicRelation.html
HomographicRelationExample.html

### References

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