the diameter of the circle goes through the center of the circle and is equal to 2 times the radius.
it is also defined as a chord that goes through the center of the circle.
any other chord that goes through the center of the circle is also a diameter, by definition.
all other chords, other than diameters, therefore have to not go through the center of the circle.
you can take any of these chords and form a triangle from them by connecting their end points to the center of the circle.
what is formed is an isosceles triangle.
you can drop a perpendicular from the vertex of this isosceles triangle and form two congruent right triangles.
the hypotenuse of these triangles is the radius of the circle.
one leg of each of these triangles is half the chord they were created from, and the other leg of each of these triangle is equal to the altitude of those triangles.
by the pythagorean formula, the hypotenuse squared is equal to the sum of each leg squared.
this means that the hypotenuse squared of each of these triangles is equal to the altitude squared plus half the chord length squared.
since the hypotenuse of each of these triangles is equal to the radius of the circle, this means that the radius squared is equal to the altitude squared plus half the chord length squared.
this means that, as long as the altitude squared is not equal to 0, the radius squared is greater than half the chord length squared.
this means that the radius is greater than half the chord length.
since the radius is equal to half the diameter, this means that half the diameter squared is greater than half the chord squared.
this means that the diameter is greater than the chord.
a picture of what i am describing to you is shown below:
in this diagram, the chord is represented by c.
the altitude is represented by a.
the diameter is represented by d.
the radius is represented by r.
the altitude divides the chord into 2 equal parts, each of which is labeled c/2.
the right triangles formed have a hypotenuse that is equal to the length of the radius.
these are labeled r.
by the pythagorean formula, r^2 = a^2 + (c/2)^2
as long as a^2 is greater than 0, r^2 has to be be greater than (c/2)^2
this means that r > (c/2)
since r is equal to d/2, this means that d/2 > c/2.
this means that d > c.
this holds true for any chord that does not go through the center of the circle.
this explains why the diameter is the longest chord in the circle.
this may even qualify as a proof.
