2006 Cyprus Seniors Provincial/2nd grade/Problem 2
Problem
Let $\Alpha, \Beta, \Gamma$ (Error compiling LaTeX. Unknown error_msg) be consecutive points on a straight line 
. We construct equilateral triangles $\Alpha\Beta\Delta$ (Error compiling LaTeX. Unknown error_msg) and $\Beta\Gamma\Epsilon$ (Error compiling LaTeX. Unknown error_msg) to the same side of .
a) Prove that $\angle\Alpha\Epsilon\Beta = \angle\Delta\Gamma\Beta$ (Error compiling LaTeX. Unknown error_msg)
b) If 
is the distance of 
form 
and 
is the distance of 
form $\Alpha\Gamma$ (Error compiling LaTeX. Unknown error_msg) prove that
$\frac{x_{1}}{x_{2}} = \frac{Area(\Alpha\Gamma\Delta)}{Area(\Alpha\Gamma\Epsilon)} = \frac{\Alpha\Beta}{\Beta\Gamma}$ (Error compiling LaTeX. Unknown error_msg)
