Some visuals are so obvious that you would think proofs are not needed. But then trying to proof them rigouresly is a whole other kettle of fish.
I was stumped by the following puzzle I made for myself: (really it is no homework question)
How do you proof that the axis of an ellipse are perpendicular?
Yes you can see it, it is obvious but seeing in itself is no proof.
Yes you cannot construct a countermodel, but again that is no proof.
I am really stumped with this one, it is so obvious, and easy to see, but a proof?
As definition of an ellipse I want to use: (reused from Wikipedia)
An ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the points is a constant.
As definition of the axis i want to use: (all made up by myself, so maybe incorrect, the second one, defining the minor axis, was a real struggle ;)
The first axis of the ellipse is the line containing the longest segment possible between two points on the ellipse.
The second axis of the ellipse is the line containing the midpoint of the two focal points and the shortest segment possible between two points on the ellipse,