Drawing all chords between six points on a circle, prove that only one triangle is formed in the circle's interior.

To get an interior triangle, you need three chords that form the sides of the triangle. Any two of those chords intersect to form a vertex of the interior triangle. Obviously two intersecting chords cannot share an endpoint, so the six points on the circle must each be the endpoint of one of the three chords forming the triangle.

If the 6 points on the circle are labelled A to F in order, then you must connect A to D, B to E, C to F to get the chords that form the sides of the triangle. Any other choice won't work because every chord must intersect the two others (to get the two vertices on that side of the triangle) and therefore has two endpoints on either side. This pairing AD, BE, CF is unique, so there is a unique interior triangle.