Difference between revisions of "2006 Cyprus Seniors Provincial/2nd grade/Problem 2"
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| − | + | Let <math>\Alpha, \Beta, \Gamma</math> be consecutive points on a straight line <math>(\epsilon)</math>. We construct equilateral triangles <math>\Alpha\Beta\Delta</math> and <math>\Beta\Gamma\Epsilon</math> to the same side of <math>(\epsilon)</math>. | |
a) Prove that <math>\angle\Alpha\Epsilon\Beta = \angle\Delta\Gamma\Beta</math> | a) Prove that <math>\angle\Alpha\Epsilon\Beta = \angle\Delta\Gamma\Beta</math> | ||
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b) If <math>x_{1}</math> is the distance of <math>A</math> form <math>\Gamma\Delta</math> and <math>x_{2}</math> is the distance of <math>\Gamma</math> form <math>\Alpha\Gamma</math> prove that | b) If <math>x_{1}</math> is the distance of <math>A</math> form <math>\Gamma\Delta</math> and <math>x_{2}</math> is the distance of <math>\Gamma</math> form <math>\Alpha\Gamma</math> prove that | ||
| − | <math>\frac{x_{1}}{x_{2}} = \frac{Area(\Alpha\Gamma\Delta)}{Area(\Alpha\Gamma\Epsilon)} = \frac{\Alpha\Beta}{\Beta\Gamma}</math> | + | <math>\frac{x_{1}}{x_{2}} = \frac{Area(\Alpha\Gamma\Delta)}{Area(\Alpha\Gamma\Epsilon)} = \frac{\Alpha\Beta}{\Beta\Gamma}</math>. |
Latest revision as of 09:54, 11 November 2006
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).
