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:

angle inscribed in 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.)

many angles inscribed in a semicircle

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!


rotate the cemicircle 180 degrees about the center of the circle

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.)

Related material

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
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