State Ptolemys Theorem you may paraphrase and explain its us

State Ptolemy’s Theorem (you may paraphrase), and explain its use in determining chords of circles for various angles.

Solution

Ptolemy\'s theorem states that for a cyclic quadrialteral ABCD; the product of its length of diagonals is equal to the sum of the products of the length of opposite sides;
For instnace; in a cyclic quadrilateral ABCD, AC*BD= AB*CD + BC*AD;

Since AB,BC,CD,DA,AC,BD are all chords of a circle (as A,B,C,D lie on a single circle), this theorem can be used to find the length of the sides. If angles are given, then the proportion of the lengths of the chords too can be determined.

For instance, if ABC is an equilateral triangle then AB=BC=CA =a (say) and let\'s take a fourth point D that lies on its circum circle, then A,B,C and D  are verticles of a quadrilateral;
Applying Ptolemy\'s theorem:
AC*BD= AB*CD+ AD*BC;
a*BD= a*CD+ AD*a

which is : BD= CD+AD; This can be used to find the length of the sides and we can state that the length of the fourth point from its farthest point is the sum of the length of the fourth point from its two nearest points;