My first thought was that a circle is an ellipse and you can always circumscribe a triangle with a circle. But your point is given any cone you can always place the triangle in the cone so that all three points line on the cone. Your suggestion shows there is a corresponding ellipse.
To see you can always place the triangle so, imagine holdings an ice cream code with vertex down on one hand and the triangle in the other. Drop the triangle into the cone. It is easy (for me) to see the triangle will come to rest with the three points on the cone (possibly with one corner of the triangle at the apex of the cone).
in article kt1cr3f306ck@legacy, Soroban at firstname.lastname@example.org wrote on 12/31/02 11:48 AM:
> Absolutely right, Ulli! > > In the past, I have gone through an elaborate and intricate proof > then have some colleague say, "No, no! You simply PROJECT the > image on a wall and..." -- embarrassing! > > Anyway, this "vision" has been bouncing around in my brain: > Select any three points on the surface of a cone. Pass a plane > through the three points -- and there is a circumscribing ellipse. > > There should be a way to generalize this approach: > "Given the three points (triangle), locate the appropriate plane" > (but please don't ask me for the algorithm). >