Difference between revisions of "Max's Theorem"
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'''Max's Theorem''' is a relationship that holds between circles and chords that lie on the circle. | '''Max's Theorem''' is a relationship that holds between circles and chords that lie on the circle. | ||
| − | = Theorem = | + | == 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 <math>O</math>, for a chord <math>AB</math> on the circle, <math>\overline {AO} = \overline {BO}</math>. | 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 <math>O</math>, for a chord <math>AB</math> on the circle, <math>\overline {AO} = \overline {BO}</math>. | ||
== Proof == | == 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 (AI), 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.''' | ||
Revision as of 19:58, 29 December 2024
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 (AI), 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.
