Diagonals in a Cyclic Quadrilateral

In a cyclic quadrilateral ABCD the ratio of the diagonals equals the ratio of the sums of products of the sides that share the diagonals' end points. In other words,

(1) AC / BD = (AB·AD + BC·CD) / (AB·BC + AD·CD).

Proof

Triangles PAD and PBC are similar, so that

PA/PB = AD/BC = PD/PC,

which can be also written as

(2) AB·AD/PA = AB·BC/PB, and
BC·CD/PC = AD·CD/PD,

In the same manner, the similarity of triangles PAB and PDC implies

(3) AB·AD/PA = AD·CD/PD.

which shows that four expressions

(4) AB·AD/PA, AB·BC/PB, BC·CD/PC, and AD·CD/PD

are all equal. (1) follows by combining the first and the thrid terms and also the second and the fourth.

References

  1. J. Hadamard, Le Géométrie Élémentaire, v. 1, 240 (l'édition onzième)

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