How many right-angled triangles can be formed with a diameter of the circle as base and with third vertex on the circumference of the circle? How many of them are right isosceles triangles?

Infinite (as the diameter subtends a 90° angle with any point on the circumference and there is no limit to the number of points on the circumference) as shown.

No. of right isosceles triangles = 2 (Triangles ABC and ABD as shown).