Cyclic pentagon and product of distances
Source: Romania TST 2009. Day 2. Problem 3.
April 21, 2009
conicsanalytic geometrygeometrycircumcircletrigonometry
Problem Statement
Prove that pentagon is cyclic if and only if
\mathrm{d(}E,AB\mathrm{)}\cdot \mathrm{d(}E,CD\mathrm{)} \equal{} \mathrm{d(}E,AC\mathrm{)}\cdot \mathrm{d(}E,BD\mathrm{)} \equal{} \mathrm{d(}E,AD\mathrm{)}\cdot \mathrm{d(}E,BC\mathrm{)}
where denotes the distance from point ot the line .