Thales' theorem

Draw a circle, run a diameter from $A$ to $B$ and put a point $C$ anywhere on the upper arc. Join $C$ to $A$ and to $B$ and a triangle appears. Now measure the angle at $C$: it is always exactly 90°, no matter where on the arc you placed $C$. This discovery is credited to Thales of Miletus, it is about 2600 years old and still the fastest route to a clean right angle.

What the theorem actually says

More precisely: if the segment $AB$ is a diameter and $C$ lies on the circle but neither on $A$ nor on $B$, then the triangle $ABC$ is right-angled, and the right angle sits at $C$. The diameter $AB$ is therefore automatically the longest side, the hypotenuse. The two shorter sides $AC$ and $BC$ are called the legs. In this role the circle is called the Thales circle. Both conditions matter: $AB$ really has to pass through the centre, and $C$ has to sit exactly on the arc. So in a circle of diameter 10 cm every such triangle has a hypotenuse of 10 cm, only the two legs change depending on where $C$ sits.

Why it always works

Call the centre $M$. Because $A$, $B$ and $C$ all lie on the circle, the segments $MA$, $MB$ and $MC$ are equally long, they are all radii. So the triangle $AMC$ is isosceles, which means the angles at $A$ and at $C$ are equal, call them $\alpha$. In the same way $BMC$ is isosceles, and there the two equal angles are called $\beta$. In the big triangle $ABC$ the angle at $A$ is $\alpha$, the angle at $B$ is $\beta$, and at $C$ the two meet, so there it is $\alpha + \beta$. Since all three angles together make 180°, only half of that is left for $\alpha + \beta$.

Building right angles, finding centres

Above all the theorem is a tool. First, it lets you construct right angles: draw a segment, halve it, swing a circle around the midpoint through both end points, and every point of the arc hands you an exact right angle, with no set square at all. Second, the converse holds: if a triangle has a right angle at $C$, then $C$ lies on the circle over the hypotenuse. The midpoint of the hypotenuse is therefore the same distance from all three corners, namely half the hypotenuse. For a hypotenuse of 13 cm that is 6.5 cm. If you only know the two legs $a$ and $b$ instead of the hypotenuse, you get the hypotenuse $c$ first from Pythagoras' theorem, $a^2 + b^2 = c^2$. Third, this is how you find the centre of a round table top: place the right angle of a set square with its corner on the rim, mark the two spots where its arms cross the rim, and join them. That line is a diameter. Repeat it somewhere else, and the two diameters cross exactly at the centre.