Thales' theorem is a fundamental result in geometry that states:

If $A$, $B$, and $C$ are points on a circle where $AB$ is a diameter of the circle, then the angle $\angle ACB$ is a right angle (90°).

Explanation:

• Consider a circle with a diameter $AB$.
• Let $C$ be any point on the circumference of the circle (but not on the diameter $AB$).
• Then, the angle $\angle ACB$ formed by drawing lines $AC$ and $BC$ to the endpoints of the diameter is always a right angle.

Proof Outline:

1. Let $O$ be the center of the circle, making $OA = OB = OC = r$ (where $r$ is the radius).
2. Triangle $OAC$ and triangle $OBC$ are isosceles because $OA = OC$ and $OB = OC$.
3. The angles at the center are $\angle AOC$ and $\angle BOC$.
4. $\angle AOC + \angle BOC = 180^\circ$ since $AB$ is a straight line through $O$.
5. Since $\angle AOC = \angle BOC$, each of these angles is $90^\circ$, making $\angle ACB = 90^\circ$.

Applications:

• Thales' theorem is widely used in constructing perpendicular lines and understanding the properties of circles.
• It also forms