conic sections
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.
PERHAPS THIS HELPS
I've been thinking about your problem for a while. I wouldn't say that I could solve it. However, I have an idea that I would like to share with you.
You certainly remember the Dandelin spheres which may have something to do with your conjecture. The following figure depicts two cones (a black one and a grey one). These cones share the vertical axis and a Dandelin sphere. Tangent to this sphere there is a blue plane whose edge can be seen in the figure:
The blue plane and the black cone intersect in a parabola. Also, the blue plane intersects with the other cone in a hyperbola. The parabola's focus and one of the foci of the hyperbola are common.
My conjecture is that this is how to imagine your couple of parabola and hyperbola with a common focus.
Then, I don't know how to proceed.
HINT.
From any point $A$ of the hyperbola, in the branch intersecting the parabola but external to it (see diagram), draw tangents $AD$ and $AF$ to the parabola, which intersect the other branch of the hyperbola at $B$ and $C$. Check that line $BC$ is also tangent to the parabola.
Best Answer
Short Proof:
You need two lemmas:
The foot of the perpendicular from the focus to a tangent of the parabola lies on the tangent at the vertex.That means the feet of the perpendicular to the three sides of the triangle formed by the tangents lie on a straight line, called the Simson Line which leads us to use
Simson-Wallace Theorem: The Feet of the perpendiculars from a point to the sides of a triangle are collinear iff the point lies on the circumcircle.
It now follows that focus lies on the circumcircle