Geometry Problems

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My Questions are:

$\bullet $ Find the minimum perimeter of triangle XYZ.

$\bullet$ Find the maximum perimeter of triangle XYZ.

$\bullet$ Find the minimum perimeter of triangle XYZ when the outer triangle has maximum perimeter.

$\bullet$ Find the maximum perimeter of triangle XYZ when the outer triangle has maximum perimeter.

$\bullet$ Find the minimum perimeter of triangle XYZ when the outer triangle has minimum perimeter.

$\bullet$ Find the maximum perimeter of triangle XYZ when the outer triangle has minimum perimeter.

Details and assumptions (In all cases)

$\bullet$ Inner triangle XYZ must touches all sides outer triangle ABC

$\bullet$ Outer triangle is an acute angled triangle.

$\bullet$ Circum-radius of outer Triangle is R=1

I posted a similar question earlier which is was a blunder. Since when I trying to answer the question which was asked by my friend that is:

" What is condition for minimum possible perimeter of inner triangle XYZ which inscribed in the outer triangle ABC when there is no constraint ?? "

Then I discover that XYZ must be orthic triangle and he tells me that I'm correct.

So my next work is that to calculate that minimum perimeter. But I'm unable to solve it at my own. So I posted a Note on this question

But from that question my curiosity is increased and I decided that if I fixed 'R', then the perimeter of Triangle XYZ should be maximum if it is an orthic triangle By Jenson's Inequality. So I posted that question

Which is later on found to be wrong.

These are such question that are coming in my mind. Please try to solve all of them and then post your solutions please!

And If you have any more such type of questions then please share with us.

Thanks!

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Markdown	Appears as
`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
- bulleted - list	bulleted list
1. numbered 2. list	numbered list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1 paragraph 2	paragraph 1 paragraph 2
`[example link](https://brilliant.org)`	example link
`> This is a quote`	This is a quote
# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"	# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"

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}`	$\boxed{123}$

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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}`	$\boxed{123}$