MHB Minimizing Isosceles triangle with a circle inscribed

[MarkFL]

So how would we compute the maximum area of the triangle?

If $b$ is unbounded, then so is the area of the triangle. It has no real maximum, we can make it as large as we want. :D

 

[Dethrone]

I found another solution to this problem, but very similar.

View attachment 3007

Can someone explain how that's similar triangles? I can kind-of visualize it if you flip it...

 

[MarkFL]

They are both right triangles, and they share an acute angle, so we know they have the same 3 internal angles, and so they must be similar. :D

 

[I like Serena]

I found another solution to this problem, but very similar.

View attachment 3007

Can someone explain how that's similar triangles? I can kind-of visualize it if you flip it...

Yep. Flip it and you can visualize it! (Mmm)
It means that the smaller triangle can be scaled to the bigger triangle (after flipping).
And that means that the ratio of the sides next to the right angle in both triangles is the same.
In the big triangle the ratio is $y:x$, while in the small triangle the ratio is $r : \sqrt{x^2-2rx}$.Alternatively, an equilateral triangle will have an extreme area due to its symmetry. Break the symmetry in any direction and the area will change.
Since you can also make the triangle as big as you want, by choosing a basis that is either large enough, or small enough, it follows that the equilateral triangle will have minimum area. (Nerd)

 

[Dethrone]

I see now that the acute angle in both right triangles are the same...but it wasn't immediately obvious. I had to work out (arbitrary) angles in my head. Anyway to quickly notice that they're similar without working out angles? I know the clue is that they're both right triangles.

 

[MarkFL]

I see now that the acute angle in both right triangles are the same...but it wasn't immediately obvious. I had to work out (arbitrary) angles in my head. Anyway to quickly notice that they're similar without working out angles? I know the clue is that they're both right triangles.

Go to the top vertex of the outer isosceles triangle, and then look at the angle up there to the right of the bisector. Do you see that both triangles share this angle? :D

 

[I like Serena]

I know the clue is that they're both right triangles.

You need 2 clues: 2 angles that are the same.
The first is that they both have a right angle.
The other is that they have an angle in common. (Whew)

 

[Dethrone]

Go to the top vertex of the outer isosceles triangle, and then look at the angle up there to the right of the bisector. Do you see that both triangles share this angle? :D

I think I know what you're saying, but not really. "and then look at the angle up there to the right of the bisector." -where does that refer?

I see what ILS is saying...both triangles share the same top angle, and the right angle, meaning that the 3rd angle must be the same! :D

 

[MarkFL]

That's what I'm saying as well. :D

 

[I like Serena]

That's what I'm saying as well. :D

Did you really need the bisector of one of the interior angles, identifying the meridian, which is also the altitude, and the orthocenter, to explain how the other interior angles relate between the non-oblique triangles? (Rofl)

 

[Dethrone]

Did you really need the bisector of one of the interior angles, identifying the meridian, which is also the altitude, and the orthocenter, to explain how the other interior angles relate between the non-oblique triangles? (Rofl)

Explain in english? I really want to understand those terms. (Nerd)

 

[I like Serena]

Explain in english? I really want to understand those terms. (Nerd)

That is English!
I'm only looking them up because English isn't my native language.
In my language I would refer to the bisectrice, zwaartelijn, hoogtelijn, respectively middelloodlijn. But yeah, then I would have to explain in English. (Giggle)

 

[Dethrone]

It was a joke, :D. But I want to understand what you wrote.

Spoiler

I like your new title, ILS, but I do think it should be bolded in color. Btw, Mark, you're the best ADMIN EVER :D :D :D

 

[MarkFL]

Did you really need the bisector of one of the interior angles, identifying the meridian, which is also the altitude, and the orthocenter, to explain how the other interior angles relate between the non-oblique triangles? (Rofl)

Yes...yes I did. (Poolparty)

 

[I like Serena]

It was a joke, :D. But I want to understand what you wrote.

It's merely a set of definitions that applies to triangles.
If you don't know them yet, you're likely to learn them soon. ;)

A bisector is the line that divides an angle in 2 equal parts.
An altitude of an angle is the line that is perpendicular to the opposing side.
A meridian divides the opposing side into 2 equal parts.
An orthocenter is a perpendicular line that divides a side into 2 equal parts.

One of the amazing facts of geometry is that each of those concepts identify a unique point in a triangle.
That is, the 3 bisectors of a triangle meet in a single point. (Nerd)

 

[I like Serena]

Spoiler

I like your new title, ILS, but I do think it should be bolded in color. Btw, Mark, you're the best ADMIN EVER :D :D :D

Spoiler

I can't. Only admins can. (Doh)

 

[Dethrone]

That's kind of weird. Bisector is what divides an angle in two, whereas the perpendicular bisector is what divides a line in two, while being perpendicular to the line. One refers to the angle being cut in two, one refers to the line being cut in two. I always thought bisector was what cut a line segment in two...:(

 

[I like Serena]

That's kind of weird. Bisector is what divides an angle in two, whereas the perpendicular bisector is what divides a line in two, while being perpendicular to the line. One refers to the angle being cut in two, one refers to the line being cut in two. I always thought bisector was what cut a line segment in two...:(

From latin bi means two and sect means divide.
To bisect merely means to divide something into 2 equal parts. (Nerd)

In a triangle that is either the bisector of an angle, or the orthocenter or bisector of a side.
In numerical mathematics the bisection algorithm is an algorithm to find the zeroes of a function by dividing the interval surrounding a zero repeatedly into 2 equal size intervals. (Angel)

 

[MarkFL]

I may not be using the term correctly, but when a plane figure has bilateral symmetry, I will refer to the axis of symmetry as that object's bisector. :D

 

[Dethrone]

Thanks! I think I'm starting to remember this now...finding the circumcenter, orthocenter, or centroid of a triangle from grade 10.

 

[I like Serena]

Thanks! I think I'm starting to remember this now...finding the circumcenter, orthocenter, or centroid of a triangle from grade 10.

Ah! You did learn! You just conveniently forgot about it, presumably being bored by a teacher that spoke too fast, or with an accent, or some such (or you were just joking). (Smirk)

 

[MarkFL]

Ah! You did learn! You just conveniently forgot about it, presumably being bored by a teacher that spoke too fast, or with an accent, or some such. (Smirk)

Wow...I am rolling here! (Rofl)(Sweating)(Clapping)