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                           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.)

                              [tbow_sh]

Related material
Read more...

                 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

                              [tbow_sh]

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