Equal sums of surfaces












1












$begingroup$


The inner points $X$ and $Y$ in the convex quadrilateral $ABCD$ are such that $angle ABX=angle CBY$, $angle BCY=angle DCX$, $angle CDY=angle ADX$ and $angle DAY=angle BAX$.
Prove that $$S^{}_{AXB}+S^{}_{CXD}=S^{}_{AYD}+S^{}_{BYC}$$






euclidean-geometry







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$endgroup$












  • $begingroup$
    What have you tried so far?
    $endgroup$
    – Larry
    Jan 6 at 13:26
















1












$begingroup$


The inner points $X$ and $Y$ in the convex quadrilateral $ABCD$ are such that $angle ABX=angle CBY$, $angle BCY=angle DCX$, $angle CDY=angle ADX$ and $angle DAY=angle BAX$.
Prove that $$S^{}_{AXB}+S^{}_{CXD}=S^{}_{AYD}+S^{}_{BYC}$$






euclidean-geometry







share|cite|improve this question











$endgroup$












  • $begingroup$
    What have you tried so far?
    $endgroup$
    – Larry
    Jan 6 at 13:26














1












1








1


1



$begingroup$


The inner points $X$ and $Y$ in the convex quadrilateral $ABCD$ are such that $angle ABX=angle CBY$, $angle BCY=angle DCX$, $angle CDY=angle ADX$ and $angle DAY=angle BAX$.
Prove that $$S^{}_{AXB}+S^{}_{CXD}=S^{}_{AYD}+S^{}_{BYC}$$






euclidean-geometry







share|cite|improve this question











$endgroup$




The inner points $X$ and $Y$ in the convex quadrilateral $ABCD$ are such that $angle ABX=angle CBY$, $angle BCY=angle DCX$, $angle CDY=angle ADX$ and $angle DAY=angle BAX$.
Prove that $$S^{}_{AXB}+S^{}_{CXD}=S^{}_{AYD}+S^{}_{BYC}$$






euclidean-geometry




euclidean-geometry






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Jan 6 at 13:41









greedoid

38.7k114797




38.7k114797










asked Jan 6 at 13:17









J. HoltanJ. Holtan

62




62












  • $begingroup$
    What have you tried so far?
    $endgroup$
    – Larry
    Jan 6 at 13:26


















  • $begingroup$
    What have you tried so far?
    $endgroup$
    – Larry
    Jan 6 at 13:26
















$begingroup$
What have you tried so far?
$endgroup$
– Larry
Jan 6 at 13:26




$begingroup$
What have you tried so far?
$endgroup$
– Larry
Jan 6 at 13:26










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