Difference between revisions of "2016 OIM Problems/Problem 5"
| Line 1: | Line 1: | ||
== Problem == | == Problem == | ||
The circles <math>\Gamma_1</math> and <math>\Gamma_2</math> intersect at two different points <math>A</math> and <math>K</math>. The tangent | The circles <math>\Gamma_1</math> and <math>\Gamma_2</math> intersect at two different points <math>A</math> and <math>K</math>. The tangent | ||
| − | common to <math>\Gamma_1</math> and <math>\Gamma_2</math> closest to <math>K</math> touches <math>\Gamma_1</math> at <math>B</math> and <math>\Gamma_2</math> at <math>C</math>. | + | common to <math>\Gamma_1</math> and <math>\Gamma_2</math> closest to <math>K</math> touches <math>\Gamma_1</math> at <math>B</math> and <math>\Gamma_2</math> at <math>C</math>. Let <math>P</math> be the foot of the perpendicular from <math>B</math> on <math>AC</math>, and <math>Q</math> the foot of the perpendicular from <math>C</math> on <math>AB</math>. If <math>E</math> and <math>F</math> are the symmetrical points of <math>K</math> with respect to the lines <math>PQ</math> and <math>BC</math>, prove that the points <math>A, E</math> and <math>F</math> are collinear. |
| − | |||
~translated into English by Tomas Diaz. ~orders@tomasdiaz.com | ~translated into English by Tomas Diaz. ~orders@tomasdiaz.com | ||
Latest revision as of 15:01, 14 December 2023
Problem
The circles 
and 
intersect at two different points 
and 
. The tangent
common to and closest to touches at 
and at 
. Let 
be the foot of the perpendicular from on 
, and 
the foot of the perpendicular from on 
. If 
and 
are the symmetrical points of with respect to the lines 
and 
, prove that the points 
and are collinear.
~translated into English by Tomas Diaz. ~orders@tomasdiaz.com
Solution
This problem needs a solution. If you have a solution for it, please help us out by adding it.
