Difference between revisions of "1985 OIM Problems/Problem 6"
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== Problem == | == Problem == | ||
| − | Given triangle <math>ABC</math>, we consider the points <math>D</math>, <math>E</math>, and <math>F</math> of lines <math>BC</math>, <math>AC</math>, and <math>AB</math> respectively. If lines <math>AD</math>, <math>BE</math>, and <math>CF</math> all pass through the center <math>O</math> of the circumference of triangle <math>ABC</math>, which radius is <math>r</math>, | + | Given triangle <math>ABC</math>, we consider the points <math>D</math>, <math>E</math>, and <math>F</math> of lines <math>BC</math>, <math>AC</math>, and <math>AB</math> respectively. If lines <math>AD</math>, <math>BE</math>, and <math>CF</math> all pass through the center <math>O</math> of the circumference of triangle <math>ABC</math>, which radius is <math>r</math>, prove: |
<cmath>\frac{1}{AD}+\frac{1}{BE}+\frac{1}{CE}=\frac{2}{r}</cmath> | <cmath>\frac{1}{AD}+\frac{1}{BE}+\frac{1}{CE}=\frac{2}{r}</cmath> | ||
| + | |||
| + | ~translated into English by Tomas Diaz. ~orders@tomasdiaz.com | ||
== Solution == | == Solution == | ||
{{solution}} | {{solution}} | ||
| + | |||
| + | == See also == | ||
| + | https://www.oma.org.ar/enunciados/ibe1.htm | ||
Latest revision as of 13:25, 13 December 2023
Problem
Given triangle 
, we consider the points 
, 
, and 
of lines 
, 
, and 
respectively. If lines 
, 
, and 
all pass through the center 
of the circumference of triangle , which radius is 
, prove:
![\[\frac{1}{AD}+\frac{1}{BE}+\frac{1}{CE}=\frac{2}{r}\]](http://latex.artofproblemsolving.com/6/1/5/615ea4eb1cd1edd46fccfb3b95cee60d42c21b01.png)
~translated into English by Tomas Diaz. ~orders@tomasdiaz.com
Solution
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