A Problem with a Heptagon

Prove that in a regular heptagon $V_1V_2\cdots V_7$ , that $\dfrac{1}{V_1V_2}=\dfrac{1}{V_1V_3}+\dfrac{1}{V_1V_4}$ .

Challenge: Prove it only using standard Geometry.

I know this problem is from a contest somewhere, but I can't find the source; it would be nice if someone can post a link to the source.

#EuclideanGeometry #Diagonals #Reciprocals #Heptagon

Easy Math Editor

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

From Ptolemy on $V_1 V_2 V_3 V_5$ , we have $V_1 V_2 \cdot V_3 V_5+V_2 V_3 \cdot V_1 V_5=V_1 V_3 \cdot V_2 V_5$ .

Because $V_2 V_5=V_1 V_4=V_1 V_5$ , $V_3 V_5=V_1 V_3$ , and $V_2 V_3=V_1 V_2$ the equation can be rewritten as $V_1 V_2 \cdot V_1 V_3+V_1 V_2 \cdot V_1 V_4=V_1 V_3 \cdot V_1 V_4$

Dividing by $V_1 V_2 \cdot V_1 V_3 \cdot V_1 V_4$ on both sides, we have $\frac{1}{V_1V_2}=\frac{1}{V_1V_3}+\frac{1}{V_1V_4}$ , as desired.

minimario minimario - 7 years, 4 months ago

Because any isosceles trapezoid is cyclic.

Bob Krueger - 7 years, 4 months ago

Alternately, since the original heptagon is cyclic.

Michael Tang - 7 years, 4 months ago

Its from my facebook page.You may see it. Its link is https://www.facebook.com/StudyingMath . And a solution using trignometry is given there.This question is somewhere in the last ones.Check it.And if u don't have fb account, Then I can post that solution here.

Dinesh Chavan - 7 years, 4 months ago

please do.i have no fb accnt

Chirayata Bhatttachharyya - 7 years, 4 months ago

Sorry for the late reply.I was engaged with my school exams.Here is a link for the imahe.If you still have any difficulties, then you can tell me. https://imageshack.com/i/141fnrp . Sorry I can't upload an image on brilliant.And let me know if you did not get any steps.And after the image It follows that 7y/4=(pi)/2 so y=2(pi)/7. therefore number of sides =7.

Dinesh Chavan - 7 years, 4 months ago

No, sorry; I'm pretty sure it is from a competition. Please do post the solution here, though.

Daniel Liu - 7 years, 4 months ago

it`s from B.MATH HONS. ADM TEST 2009 BY ISI

Chirayata Bhatttachharyya - 7 years, 4 months ago

The problems come from Indian INMO 1992. See here : Its problem 9.

Shivang Jindal - 7 years, 4 months ago

well I know that too,still it`s not the exact one.

Chirayata Bhatttachharyya - 7 years, 4 months ago

An announcement here: sorry for not posting any #CosinesGroup posts recently. I have a bunch of homework recently, so I didn't find the time to write any.

I promise that the next post will come no later than a week :)

Daniel Liu - 7 years, 4 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$