The Angle Inscribed in a Semicircle

An inscribed angle is an angle whose vertex sits on the circle and whose two sides are chords reaching to the two ends of some arc. The core rule for circles: an inscribed angle always measures half the arc it cuts off. A diameter splits the circle into two arcs of 180 degrees each. So any angle inscribed on that diameter, from any point on the rim, cuts off a 180 degree arc and must measure exactly half of that: 90 degrees. The point you pick on the circle doesn't matter. Move it anywhere along the rim and the angle stays 90, because the arc behind it never changes.

This is why not every triangle drawn inside a circle is a right triangle. The right angle only shows up when one side of the triangle happens to be a diameter. If all three corners sit on the circle but none of the sides passes through the center, none of the arcs is 180 degrees, so none of the inscribed angles is forced to 90. The special case is the diameter, nothing else.

Once you see the diameter as the hypotenuse, the rest is just the Pythagorean theorem. The diameter is always the longest side, since it's the hypotenuse of a right triangle, and the two legs run from the ends of the diameter to the point on the rim. This links straight into chord and arc problems: the same half-the-arc rule that gives you this 90 degree angle also drives inscribed angle problems that don't involve a diameter at all.