# Two Touching Ellipses - Tangents, Centres and Collinearity

Consider two ellipses, touching each other at a point (i.e. they've a common tangent at that point). It is given that they also have two more external common tangents, which when extended meet at P. The centres of the ellipses are A and B.

Prove that A, B, P are collinear.

I've attached a diagram, for the sake of clarity -

I tried using coordinate geometry for this, assuming one to be a standard ellipse (axes parallel to coordinate axes) and the other to be any touching ellipse. This method is really cumbersome - finding the equation of the second ellipse, the tangents, and then the intersection point is not at all elegant and seems very time consuming.

However, it is important to note that the combined equation of tangents from point P to both ellipses should be identical. (if that helps?)

Nevertheless, this problem can probably be solved easily using the geometrical properties of an ellipse (I'm not sure how, but seems like it'll be better than a pure coordinate geometry solution).

So, how to prove the collinearity of those 3 points? Please explain with a detailed solution.

P.S. Of course, if possible, it'll be great to know how to solve it using both methods (or more) - analytic geometry and pure geometry.

• Colinearity is an affine property and every ellipse is an affine image of a circle. Have you tried making one of the two ellipse a circle?
– amd
Commented Mar 17, 2018 at 6:50
• Can we try this without affine transformations too? I haven't tried it yet, I will. Commented Mar 17, 2018 at 6:51
• You could create a projection in which the to ellipses become circles.
– Moti
Commented Mar 17, 2018 at 21:07