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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>ABCD</math>, that has no parallel sides, is inscribed in a circle, its sides <math>DA</math>, <math>CB</math> meet at <math>E</math> and its sides <math>BA</math>, <math>CD</math> meet at <math>Z</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 DEC</math> and <math>\angle CZB</math> intersect the sides of the quadrilateral at the points <math>K , L, M ,N</math> prove that
  
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i)the bisectors intersect normally
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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>K , L, M ,N</math> are vertices of a rhombus.
  
 
== Solution ==
 
== Solution ==

Latest revision as of 23:10, 19 February 2020

Problem

A quadrilateral $ABCD$, that has no parallel sides, is inscribed in a circle, its sides $DA$, $CB$ meet at $E$ and its sides $BA$, $CD$ meet at $Z$. If the bisectors of of $\angle DEC$ and $\angle CZB$ intersect the sides of the quadrilateral at the points $K , L, M ,N$ prove that

i)The bisectors intersect normally

ii)the points $K , L, M ,N$ are vertices of a rhombus.

Solution

See also

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