Pick any two fixed points on a circle, and draw straight lines from each of them to some third point elsewhere on the circle's edge — the angle formed where those two lines meet stays exactly the same, no matter which third point you choose, as long as it stays on the same side of the circle relative to the two fixed points. That's the inscribed angle theorem, one of classical geometry's cleaner results: a claim about angles that holds true across infinitely many different positions around a circle, not just for one specific configuration.
What "inscribed" and "subtending" actually mean here
An inscribed angle is an angle formed by two chords of a circle that share a common endpoint on the circle's edge — that shared endpoint is the angle's vertex. The arc "cut off" by the angle's two other endpoints, the part of the circle's edge lying between them on the far side from the vertex, is the arc the angle is said to subtend. The inscribed angle theorem states that every inscribed angle subtending the same arc has the same measure, specifically half the measure of the central angle that subtends the same arc from the circle's centre — a relationship that holds regardless of exactly where along the remaining arc the inscribed angle's vertex happens to sit.
A special case that turns any diameter into a guaranteed right angle
A particularly clean special case follows directly from the general theorem: if the arc being subtended is a semicircle — meaning the two chord endpoints lie at opposite ends of a diameter — then the central angle subtending that arc is a straight angle of 180 degrees, and by the theorem, every inscribed angle subtending it must measure exactly half of that, 90 degrees. This means any triangle formed by a circle's diameter and a third point anywhere else on the circle is automatically a right triangle, a result known as Thales' theorem, sometimes credited as one of the earliest recorded pieces of formal deductive geometry. It's a striking demonstration of the general theorem's power: a guaranteed right angle, produced automatically by picking any point at all on a circle, as long as the base of the triangle is a diameter.
What we're still unsure about
The inscribed angle theorem and its special case, Thales' theorem, are rigorously proven results in Euclidean geometry, established with full mathematical certainty and taught consistently across geometry curricula. What's less settled, and more a matter of historical interpretation, is the precise attribution and dating of the earliest proofs — Thales' theorem is traditionally credited to the ancient Greek mathematician Thales of Miletus, but the historical record from that period is thin, and how much of the specific proof attributed to him actually originated with him, versus being organised and formalised later by other mathematicians building on earlier, less rigorously documented geometric knowledge, isn't something historians of mathematics can establish with full confidence.
This sits inside Circles & Arc Theorems, one of eight topics in Geometry, one of seven domains in Mathematics, one of seventeen subjects the app can quiz you on.