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Difference between revisions of "2006 Seniors Pancyprian/2nd grade/Problem 4"

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== Problem ==
 
== Problem ==
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A quadrilateral <math>\Alpha\Beta\Gamma\Delta</math>, that has no parallel sides, is inscribed in a circle, its sides <math>\Delta\Alpha</math>, <math>\Gamma\Beta</math> meet at <math>\Epsilon</math> and its sides <math>\Beta\Alpha</math>, <math>\Gamma\Delta</math> meet at <math>\Zeta</math>.
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A quadrilateral <math>\alpha \beta \gamma \delta</math>, that has no parallel sides, is inscribed in a circle, its sides <math>\delta \alpha</math>, <math>\gamma \beta</math> meet at <math>\epsilon</math> and its sides <math>\beta\alpha</math>, <math>\gamma\delta</math> meet at <math>\zeta</math>.
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If the bisectors of of <math>\angle\Delta\Epsilon\Gamma</math> and <math>\angle\Gamma\Zeta\Beta</math> intersect the sides of the quadrilateral at th points <math>\Kappa, \Lambda, \Mu, \Nu</math> prove that
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If the bisectors of of <math>\angle\delta\epsilon\gamma</math> and <math>\angle\gamma\zeta\beta</math> intersect the sides of the quadrilateral at the points <math>\kappa, \lambda, \mu, \nu</math> prove that
  
 
i)the bisectors intersect normally
 
i)the bisectors intersect normally
  
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ii)the points <math>\Kappa, \Lambda, \Mu, \Nu</math> are vertices of a rhombus.
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ii)the points <math>\kappa, \lambda, \mu, \nu</math> are vertices of a rhombus.
  
 
== Solution ==
 
== Solution ==

Revision as of 23:04, 19 February 2020

Problem

A quadrilateral $\alpha \beta \gamma \delta$, that has no parallel sides, is inscribed in a circle, its sides $\delta \alpha$, $\gamma \beta$ meet at $\epsilon$ and its sides $\beta\alpha$, $\gamma\delta$ meet at $\zeta$. If the bisectors of of $\angle\delta\epsilon\gamma$ and $\angle\gamma\zeta\beta$ intersect the sides of the quadrilateral at the points $\kappa, \lambda, \mu, \nu$ prove that

i)the bisectors intersect normally

ii)the points $\kappa, \lambda, \mu, \nu$ are vertices of a rhombus.

Solution

See also

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