There are many ways to prove this theorem. One of the most classic proofs is as follows:
We know as all of them are the radii of the circle. Hence, and because angles opposite to equal sides are equal.
Then we have
There's another way of proving this, which is based on another theorem called the alternate segment theorem which states that
Alternate segment theorem:
The angle subtended by a chord (or two radii) at the center of a circle is two times the angle subtended by it on the remaining part of the circle.
Let us now try to prove Thales' theorem with the help of the above theorem.
According to the angle segment theorem, we have the following diagram:
Let us say that the radii and form the diameter, then the figure would look like this:
is the diameter of a semi-circle.
is a point on the circumference.
What is the measure of angle
The longest chord in a circle is its diameter.
Let a circle with center have a chord . Join and , and let be a point on such that .
We know that the perpendicular from the center of a circle to a chord bisects the chord. Then .
Let the radius be , the length of chord be , and the perpendicular distance between the center and the chord be .
Now, by the Pythagorean theorem, in
So, we need to find the maximum possible value of , right? We know that a function is maximized when its derivative is equal to zero. Then, since radius is constant, is maximized when
Then, is maximized when , implying that the maximum value of is
True or false?
The length of the largest chord possible in a circle is always equal to twice its radius.