Equilateral triangle inscribed in a ellipse **"Given any point on a ellipse, is it always possible to inscribe an equilateral triangle, with a vertex coincident with that point, in the ellipse?"** I thought I could use _analytical geometry_ , but when I first come up with a line-long equation I gave it up. Do you think there are faster, and more _elegant_ ways to solve this problem?

I have an idea, but a rigorous proof based on this idea may be a bit tough. Nevertheless, perhaps you find it useful.

For each $r>0$, consider the circle centered on the given point $P$ of the ellipse with radius $r$.

If the circle intersects the ellipse two points, they will be at a distance $d(r)$. We define $f(r)=d(r)/r$. If the circle intersects the ellipse more than two points $A_1,\ldots\,A_n$ take $A_i, A_j$ such that the arc $A_iPA_j$ doesn't cointain any other $A_k$.

If the circle is tangent to the ellipse (being the ellipse within the circle), define $f(r)=0$.

If the circle and the ellipse don't share any points, $f$ remains undefined.

It seems that for small enough $r$, the triangle defined by the given point and the intersection has an obtuse angle, so $f(r)>1$. It also seems that $f$ is continuous where we have defined it. So there will be some $r$ such that $f(r)=1$.