Suppose that we have a convex polygon with some red sides and some blue sides. Suppose that it has the property that there does not exist two red sides that are adjacent, but the total lengths of the red sides combined is more than the total lengths of the blue sides.

Prove that it is impossible to inscribed within the polygon a circle.

No vote yet

1 vote

×

Problem Loading...

Note Loading...

Set Loading...

Easy Math Editor

`*italics*`

or`_italics_`

italics`**bold**`

or`__bold__`

boldNote: you must add a full line of space before and after lists for them to show up correctlyparagraph 1

paragraph 2

`[example link](https://brilliant.org)`

`> This is a quote`

Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.`2 \times 3`

`2^{34}`

`a_{i-1}`

`\frac{2}{3}`

`\sqrt{2}`

`\sum_{i=1}^3`

`\sin \theta`

`\boxed{123}`

## Comments

Sort by:

TopNewestAll right it has been long enough for me to post a solution.

First, suppose that we can inscribe a circle in the polygon. Note that the sum of the lengths of the blue sides is at least the sum of the red sides because of Power of a Point on each vertex. This contradicts our restriction that the length of the red sides is more than the length of the blue sides, so we are done. \(\Box\)

Log in to reply

i think putting a diagram may help me somehow....i am lost halfway in the question....however indeed a nice question...

Log in to reply

Whats the prove??

Log in to reply

You will have to find that out yourself.

If it's long enough I might post one. But for now, post your ideas.

Log in to reply

Yes it is impossible to inscribe a circle within this polygon because its impossible to make such a polygon

If u are trying to make a polygon of even number sides, number of red sticks will be equal to the number if blue sticks

If u are trying to make a polygon of odd number of sides, number of blue sticks will always be greater than number of red sticks

Its my opinion please correct me if I am wrong

Log in to reply

Your argument fails for irregular polygons.

Log in to reply

I could unprove it by doing a small Google search.

Log in to reply

Can you explain how that unproves it? I don't see it.

Log in to reply