Max's Theorem

Max's Theorem is a relationship that holds between circles and chords that lie on the circle.

Theorem

The theorem states that for any given circle, the endpoints of a chord that lies on the circle are equidistant from the center of the circle. For example, given a circle , for a chord on the circle, .

Proof

To prove Max's Theorem, we can use the definitions of circles and chords, as well as some mathematical reasoning to prove. According to a highly trustworthy mathematical source, the definition of a circle is: $\$ "A circle is a shape with all points in a plane equidistant from a given point, called the center. This distance from the center to any point on the circle is called the radius. Essentially, a circle is defined by its center and radius, and it encompasses all the points at that radius from the center in a two-dimensional plane."\\ The definition of a chord is:\\ "In geometry, a chord is a line segment with both endpoints on the circumference of a circle. Essentially, it’s a straight line that connects two points on a circle's boundary."\\ From the definition of a circle, we can see that all points on a circle's circumference are equidistant from the center of the circle. Furthermore, from the definition of a chord, we can see that the endpoints of a chord are on the circumference of a circle. Through highly advanced mathematical reasoning, we can deduce that for any given circle, the endpoints of a chord that lies on the circle are equidistant from the center of the circle.