## Thales' Theorem

A useful theorem attributed to the probably first mathematician ever, Thales of Miletus (ca. 624 BC-ca. 546 BC) is one of most elementary and intuitive ever stated.

An angle inscribed in a semicircle is right, and vice versa.

A seventh-grader student of Paul Lockhart found a beautiful proof. The following is an excerpt from Paul's online article:

As a more serious example, let's take the case of a triangle inside a semicircle: Now the beautiful truth about this pattern is that no matter where on the circle you place the tip of the triangle, it always forms a nice right angle. (I have no objection to a term like "right angle" if it is relevant to the problem and makes it easier to discuss. It's not terminology itself that I object to, it's pointless unnecessary terminology. In any case, I would be happy to use "corner" or even "pigpen" if a student preferred.) Here is a case where our intuition is somewhat in doubt. It's not at all clear that this should be true; it even seems unlikely - shouldn't the angle change if I move the tip? What we have here is a fantastic math problem! Is it true? If so, why is it true? What a great project! What a terrific opportunity to exercise one's ingenuity and imagination!

... Take the triangle and rotate it around so it makes a foursided box inside the circle. Since the triangle got turned completely around, the sides of the box must be parallel, so it makes a parallelogram. But it can't be a slanted box because both of its diagonals are diameters of the circle, so they're equal, which means it must be an actual rectangle. That's why the corner is always a right angle.

...

(AB: note that) Since the triangle got rotated halfway around the circle, the tip must end up exactly opposite from where it started. That's why the diagonal of the box is a diameter.

(There is dynamic illustration of Thales' Theorem on a separate page.) ### Angles in Circle

• Angle Subtended by a Diameter
• Inscribed Angles
• Inscribed and Central Angles in a Circle
• Munching on Inscribed Angles
• Sangaku with Angle between a Tangent and a Chord
• Secant Angles in a Circle
• Secant Angles in a Circle II
• 