Thales' theorem
/ THAY-leez /
Pin the two ends of a string to the two ends of a circle's diameter, and walk a pencil along the rim keeping the string taut at the pencil — the two string-segments always meet at a perfect right angle, wherever on the upper rim the pencil sits. That clean fact, that any point on a semicircle 'sees' the diameter at 90 degrees, is Thales' theorem, named for the early Greek geometer Thales of Miletus.
Precisely: if A and B are the endpoints of a diameter of a circle, and C is any other point on the circle, then the angle ACB is a right angle (90 degrees). It is the special case of the inscribed angle theorem in which the intercepted arc is a semicircle: a semicircular arc measures 180 degrees, so the inscribed angle standing on it is half of that, 90 degrees. The converse is just as useful: if angle ACB is a right angle, then C lies on the circle with diameter AB — so the right-angle vertices of all right triangles on a fixed hypotenuse trace out a circle whose diameter is that hypotenuse.
Thales' theorem is a everyday tool: it constructs a right angle without a set square (draw a semicircle on any segment as diameter, pick any rim point), it locates the centre of a circle (two such right angles), and it shows why the midpoint of a right triangle's hypotenuse is equidistant from all three vertices (it is the circumcentre). A caution on attribution and scope: the 'theorem' is the half-angle special case only — it does NOT say every inscribed angle is 90 degrees, only those standing on a diameter. And the name 'Thales' theorem' is also attached, confusingly, to a different result about parallel lines cutting proportional segments (the intercept theorem); here we mean strictly the semicircle right-angle result.
Let AB be a diameter of a circle and C any other point on the circle. Then triangle ACB is right-angled at C. Conversely, given a right triangle with hypotenuse 10, its right-angle vertex lies on the circle of diameter 10, whose centre — the hypotenuse's midpoint — is 5 from each of the three vertices.
Any point on a semicircle sees the diameter at a right angle.
Thales' theorem applies only to angles standing on a diameter (a semicircular arc) — not to every inscribed angle. The same name is also used for an unrelated intercept (parallel-lines) theorem, so context matters.