Can you fit every single permutation of a triangle inside a circle with all three points touching the edges of the circle

Question

I am having a heated debate with some colleagues about this. I don't think that you can, but am unsure on how to prove this whereas other people think you always can but again cannot prove it. Is there a proof out there for this?

By permutations I mean any type of triangle with varying length of side and varying angle size.

I think I have proven it cannot be done using a right angled triangle.

Answer 1

Every triangle can be circumscribed by a circle. To see this, given a triangle $ABC$, note that the centre of the circle, $O$, by definition must be equidistant from the three points. Thus to construct the circle, draw the perpendicular bisector of $AB$ as well as the perpendicular bisector of $AC$; their intersection gives you $O$.