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Source: INMO 2002 Problem 1

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Rushil

#1 h Oct 10, 2005, 5:16 AM • 2 Y

Y by Adventure10, Mango247

For a convex hexagon $ABCDEF$ in which each pair of opposite sides is unequal, consider the following statements.

( $a_1$ ) $AB$ is parallel to $DE$ . ( $a_2$ ) $AE = BD$ .

( $b_1$ ) $BC$ is parallel to $EF$ . ( $b_2$ ) $BF = CE$ .

( $c_1$ ) $CD$ is parallel to $FA$ . ( $c_2$ ) $CA = DF$ .

$(a)$ Show that if all six of these statements are true then the hexagon is cyclic.

$(b)$ Prove that, in fact, five of the six statements suffice.

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111 posts

#2 h Oct 10, 2005, 5:02 PM • 2 Y

Y by Adventure10, Mango247

Rushil wrote:

$ABCDEF$ is a convex hexagon. Consider the following statements.

(1) $AB$ is parallel to $DE$ .

(2) $BC$ is parallel to $EF$ .

(3) $CD$ is parallel to $FA$ .

(4) $AE = BD$ .

(5) $BF = CE$ .

(6) $CA = DF$ .

Show that if all six of these statements are true then the hexagon is cyclic. Prove that, in fact, five of the six statements suffice.

The result seems to be false, indeed.
Take an equilateral triangle $\triangle ABC$ , and consider its reflection $\triangle A'B'C'$ in a line paralel
to $BC$ , close to $BC$ and between $A$ and $BC$ . Then the hexagon $AB'BA'CC'$ satisfies all the
statements in the problem, but is not cyclic.

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6555 posts

#3 h Nov 13, 2005, 11:51 AM • 2 Y

Y by Adventure10, Mango247

Ociretsih is right: Not even all six statements (let alone five of them) yield that the hexagon is cyclic.

What is actually true is: If a convex hexagon ABCDEF satisfies AB || DE, BC || EF, CD || FA and AD = BE = CF, then it is cyclic. This was proven on http://www.cip.ifi.lmu.de/~grinberg/Lamoen/Support.html .

Darij

This post has been edited 1 time. Last edited by darij grinberg, Sep 13, 2009, 11:05 AM

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#4 h Mar 8, 2009, 8:03 PM • 1 Y

Y by Adventure10

Hey I just saw this question. Its a bit wrong. The correct question ( which I hope the mods will kindly replace the original one posted in the resources section.....)

For a convex hexagon $ABCDEF$ in which each pair of opposite sides is unequal, consider the following statements.

( $a_1$ ) $AB$ is parallel to $DE$ . ( $a_2$ ) $AE = BD$ .

( $b_1$ ) $BC$ is parallel to $EF$ . ( $b_2$ ) $BF = CE$ .

( $c_1$ ) $CD$ is parallel to $FA$ . ( $c_2$ ) $CA = DF$ .

$(a)$ Show that if all six of these statements are true then the hexagon is cyclic.

$(b)$ Prove that, in fact, five of the six statements suffice.

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904 posts

#5 h Mar 10, 2009, 3:31 PM • 2 Y

Y by Adventure10, Mango247

Hey can anyone answer this question....????

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904 posts

#6 h Oct 2, 2009, 4:20 PM • 2 Y

Y by Adventure10, Mango247

Rijul saini wrote:

For a convex hexagon $ABCDEF$ in which each pair of opposite sides is unequal, consider the following statements.

( $a_1$ ) $AB$ is parallel to $DE$ . ( $a_2$ ) $AE = BD$ .

( $b_1$ ) $BC$ is parallel to $EF$ . ( $b_2$ ) $BF = CE$ .

( $c_1$ ) $CD$ is parallel to $FA$ . ( $c_2$ ) $CA = DF$ .

$(a)$ Show that if all six of these statements are true then the hexagon is cyclic.

$(b)$ Prove that, in fact, five of the six statements suffice.

here is the official solution....

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18 posts

#7 h Nov 7, 2022, 9:58 PM

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We can prove that each of the parallel sides form an isosceles trapezium. Further all the diagonals are concurrent at a point this proves that it can be a cyclic.

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