TITLE: Proving that the focus of a parabola lies on the circumcentre of a triangle
QUESTION [6 upvotes]: I recently came across the fact that if a parabola touches the three sides of a triangle  then the focus of such a parabola lies on the circumcircle of the above triangle.
I tried to prove it but without much information I couldnot get where to start with .Does the above property of the parabola applies to other conic sections as well? Any help is appreciated.Thanks.

REPLY [3 votes]: Another way: Let the three lines be tangent at $t_1, t_2, t_3$ on the parabola. Then the points of intersection of these tangents, which form the vertices of the triangle are $A(at_1t_2, a(t_1+t_2))$ etc. 
You can then show that these points along with $(a,0)$ form a cyclic quadrilateral, by using slopes to show that opp angles are supplementary.