Thales's theorem

– Dante's Paradiso, Canto 13, lines 100–102Non si est dare primum motum esse, Or if in semicircle can be made Triangle so that it have no right angle.

Modern scholars believe that Greek deductive geometry as found in Euclid's Elements was not developed until the 4th century BC, and any geometric knowledge Thales may have had would have been observational.

Dante Alighieri's Paradiso (canto 13, lines 101–102) refers to Thales's theorem in the course of a speech.

We will show that △ABC forms a right angle by proving that AB and BC are perpendicular — that is, the product of their slopes is equal to −1.

We calculate the slopes for AB and BC: Then we show that their product equals −1: Note the use of the Pythagorean trigonometric identity

Since lines AC and BD are parallel, likewise for AD and CB, the quadrilateral ACBD is a parallelogram.

The converse of Thales's theorem is then: the center of the circumcircle of a right triangle lies on its hypotenuse.

This proof consists of 'completing' the right triangle to form a rectangle and noticing that the center of that rectangle is equidistant from the vertices and so is the center of the circumscribing circle of the original triangle, it utilizes two facts: Let there be a right angle ∠ ABC, r a line parallel to BC passing by A, and s a line parallel to AB passing by C. Let D be the point of intersection of lines r and s. (It has not been proven that D lies on the circle.)

The quadrilateral ABCD forms a parallelogram by construction (as opposite sides are parallel).

This proof utilizes two facts: Let there be a right angle ∠ ABC and circle M with AC as a diameter.

Then we know It follows This means that A and B are equidistant from the origin, i.e. from the center of M. Since A lies on M, so does B, and the circle M is therefore the triangle's circumcircle.

The above calculations in fact establish that both directions of Thales's theorem are valid in any inner product space.