The diminishing angle paradox is the result in geometry that if a point A is equidistant from two other points B and C, where the distance between B and C is fixed, then increasing the distance of A from both points results in the diminishing size of the angle at A.

This is easy to see because an exterior angle is greater than either of the opposite interior angles. It also has the effect of causing the hypotenuse in either right angle to start coinciding with the perpendicular of AB.

If A is very far, then the points B and C become indiscernible from each other no matter how great the distance between B and C might be.

The problem is this: If we set out for A from B and then tried to return to C, how would we calculate the shortest route from A to C, since the size of the angle at A will be very close to zero?