Mark two points A and B on a circle, which fixes an arc AB. Now mark a third point P on the far side of that arc (the bigger remaining arc) and measure angle APB. What happens if you move P to a different spot on the circle and measure the angle again? Remarkably, no matter where you move P, angle APB always comes out exactly the same. This angle is called the inscribed angle on arc AB, and this property is called the inscribed angle theorem.
To understand why the inscribed angle always stays the same, you need to know its relationship with the central angle. The central angle is the angle formed at the circle's center O looking at A and B (∠AOB). According to the inscribed angle theorem, the inscribed angle on the same arc is always exactly half of the central angle. Since the central angle is a single fixed value no matter where P sits, its half — the inscribed angle — has no choice but to stay the same value too, wherever P happens to be. In the end, the fact that "the inscribed angle is constant" is simply a natural consequence of the more fundamental relationship "inscribed angle = central angle ÷ 2."
There's one especially famous special case of this property. If A and B sit exactly opposite each other across the circle's center (in other words, if segment AB is a diameter), the central angle becomes 180°. Then the inscribed angle becomes half of that, or 90°. In other words, an inscribed angle that looks at a diameter is always a right angle. This fact is also known as "Thales' theorem," named after the ancient Greek mathematician. Draw any triangle inside a semicircle, and the angle at the vertex facing the diameter always comes out to exactly 90°.
The inscribed angle theorem is a key tool for finding angles in all kinds of geometry problems involving circles. Even a shape inside a circle that looks complicated often gets much easier the moment you notice "these two angles are inscribed angles looking at the same arc." And the reverse works too — if you know an angle is 90°, you can conclude it must be looking at a diameter.
To prove this property, you use the exterior angle property of a triangle (an exterior angle equals the sum of the two non-adjacent interior angles). Draw an auxiliary line connecting the center O to point P, and you get two isosceles triangles; carefully working through the angle relationships in those triangles proves that the inscribed angle is exactly half the central angle. At this stage, it matters less that you memorize the proof perfectly and more that you get a visual feel for "why it's always exactly half."
On our activity page, change the central angle with a slider and move point P freely around the circle to check for yourself whether the inscribed angle really does stay at exactly half the central angle. Press the "Make it a diameter" button to see Thales' theorem in action right away too.