(Ptolemy's theorem). Let ABCD be a cyclic quadrilateral (a quadri- lateral whose vertices lie on a...

 

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(Ptolemy's theorem). Let ABCD be a cyclic quadrilateral (a quadri- lateral whose vertices lie on a common circle). Show that |AC||BD| = |AB||CD| + |BC||AD|. Hint. After you draw the diagonals, you should see a lot of triangles and many pairs of congruent inscribed angles. There are many ways to proceed here. The sine theorem should be useful along with some trig identities. Have patience with this one. B As we have seen, many special triplets of lines in a triangle are concurrent. The following theorem of Ceva is the most general result one can prove in this direction: (Ptolemy's theorem). Let ABCD be a cyclic quadrilateral (a quadri- lateral whose vertices lie on a common circle). Show that |AC||BD| = |AB||CD| + |BC||AD|. Hint. After you draw the diagonals, you should see a lot of triangles and many pairs of congruent inscribed angles. There are many ways to proceed here. The sine theorem should be useful along with some trig identities. Have patience with this one. B As we have seen, many special triplets of lines in a triangle are concurrent. The following theorem of Ceva is the most general result one can prove in this direction: