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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!

Note by Deepanshu Gupta
2 years, 4 months ago

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Comment deleted Oct 29, 2014

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@Deepanshu Gupta btw, is there a theorem like Routh's that gives the area of a triangle if three points are given, one on each side, and all three are connected. In this case, given the ratio that each side of ABC is divided into, what is the area of XYZ.

Rouths is that given the ratio each side of ABC is divided into, if lines are drawn to each vertex, you can find the area of the floating triangle created.

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Trevor Arashiro · 2 years, 4 months ago

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@Trevor Arashiro It is really interesting , please give me some time I want to give an try to it..! Deepanshu Gupta · 2 years, 4 months ago

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@Deepanshu Gupta Q. Find minimum perimeter of triangle XYZ ?

A. minimum is 0 because triangle ABC can be degenerate and XYZ can be degenerate as well.


Q. Find Maximum Perimeter of triangle XYZ ?

A. \(3\sqrt3\). This is because XYZ is considered to be inscribed in ABC as each vertex of XYZ approaches distance 0 from each corresponding vertex of ABC. In your solution, you provided a great proof for why the maximum perimeter of ABC was an equilateral triangle. Thus the max of XYZ=max of ABC


Q. Find Minimum Perimeter of triangle XYZ when Outer Triangle Has Maximum Perimeter ?

A. The answer to this is the previous incorrect answer to your question, since the max of ABC is \(3\sqrt3\), the minimum of XYZ is \(3\sqrt3/2\). This is the orthic triangle.


Q. Find Maximum Perimeter of Triangle XYZ when outer Triangle Has Maximum Perimeter ?

A. This is the same as your second question in this note, finding the max of XYZ.


Q. Find Minimum Perimeter of triangle XYZ when Outer Triangle Has Minimum Perimeter ?

A. This is the same as your first question in this note.


Q. Find Maximum Perimeter of triangle XYZ when Outer Triangle Has Maximum Perimeter ?

A. This is the same as your 4th question. Trevor Arashiro · 2 years, 4 months ago

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@Trevor Arashiro Awesome ..! Thanks a lot Trevor ...!!

And last question is repeated.. So I changed That . And in my views This question is meaningless since ABC has 0 minimum perimeter So in that case outer Triangle is also degenerate am I right ? Deepanshu Gupta · 2 years, 4 months ago

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@Deepanshu Gupta Its still a good problem, and don't worry, we all have blunders such as these. I am in the process of answering all of these. Trevor Arashiro · 2 years, 4 months ago

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@Trevor Arashiro You are brilliant Rama Devi · 1 year, 9 months ago

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