What is a circle?

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What is a circle?

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Source: MOP 1998

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arandomperson123

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arandomperson123

#1 h Feb 18, 2017, 10:48 PM • 2 Y

Y by Adventure10, Mango247

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Now that I have your attention, please solve this problem:

Let $ABCDEF$ be a cyclic hexagon with $AB = CD = EF$ . Prove that the intersections of $AC$ with $BD$ , of $CE$ and $DF$ , and of $EA$ and $FB$ will form a triangle similar to $BDF$

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#2 h Feb 20, 2017, 8:33 AM • 3 Y

Y by arandomperson123, Adventure10, Mango247

$X=BD$ $\cap$ $AC$ , $Y=BF$ $\cap$ $AE$ , $Z=CE$ $\cap$ $FD$ . $AE=BF$ $\Rightarrow$ $\angle$ $ACE=$ $\angle$ $BDF$ ,hence $C,X,Z,D$ are concyclic.Similarly $BXYA$ and $FYZE$ are cyclic. $\angle$ $AXZ=$ $\angle$ $ZDC=$ $\angle$ $FEC$ , $\angle$ $AXY=$ $\angle$ $ABY$ $=$ $\angle$ $AEF$ $\Rightarrow$ $\angle$ $AEC=$ $\angle$ $YXZ$ .Similarly $\angle$ $CAE=$ $XZY$ , $\angle$ $ACE=$ $\angle$ $\angle$ $XYZ$ $\Rightarrow$ $\bigtriangleup$ $XYZ$ $\cong$ $\bigtriangleup$ $\cong$ $ECA$ $\cong$ $\bigtriangleup$ $FDB$ .

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