# 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

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

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