1985 OIM Problems/Problem 6

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Problem

Given triangle $ABC$, we consider the points $D$, $E$, and $F$ of lines $BC$, $AC$, and $AB$ respectively. If lines $AD$, $BE$, and $CF$ all pass through the center $O$ of the circumference of triangle $ABC$, which radius is $r$, prove: \[\frac{1}{AD}+\frac{1}{BE}+\frac{1}{CE}=\frac{2}{r}\]

~translated into English by Tomas Diaz. ~orders@tomasdiaz.com

Solution

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See also

https://www.oma.org.ar/enunciados/ibe1.htm