How many Heronians

Geometry Level 4

How many Heronian triangles have Steiner inellipse with semi-major axis 400 and semi-minor axis 200?

2 0 3 Infinitely many None of the above 4 6 1

1 solution

Maria Kozlowska
Feb 19, 2016

Ratio of area of a triangle to area of its Steiner inellipse is always: $\frac{\triangle ABC}{\pi a b} = \frac{3\sqrt{3}}{\pi} \Rightarrow \frac{\triangle ABC}{\ a b} = 3\sqrt{3}$

For integer semiaxes there are no Heronian triangles possible.

Of course I had to click "None of the above" when I was thinking just "NONE"

Michael Mendrin - 5 years, 3 months ago

After I solved Falzoon's Ellipse I clicked View Solution instead of Submit.

Maria Kozlowska - 5 years, 3 months ago

Too bad we can't just hit the "back" button and start over in cases like this.

Michael Mendrin - 5 years, 3 months ago

@Michael Mendrin – Wait.... Falzoon is on Brilliant? Who is he/she? What is their profile account?

Pi Han Goh - 5 years, 3 months ago

@Pi Han Goh – Rita the Dog was here on Brilliant briefly, but didn't stick around. Vikram Pandya, who got me started here, does visit occasionally. Falzoon, as far as I know, never came around here, but I posted a problem based on one of his problems, in his honor. Here it is

Falzoon's Ellipse

You can see the rest of my posted problems here

Michael Mendrin Problems

What I miss about the Y!A days is that regulars talked a lot more. There doesn't seem to be as much of that here in Brilliant, even though the format is pretty Brilliant. Great LaTeX and graphics that were impossible in Y!A. I hear Yahoo! is now in deep trouble.

Michael Mendrin - 5 years, 3 months ago

@Michael Mendrin – Oh good to know. You forgot to mention the other two legends: Dragan K and gianlino. I would have mention Madhukar Daftary, but unfortunately he's gone already.

I do not know any of them personally, do you know how to contact (any of) them?

Pi Han Goh - 5 years, 3 months ago

@Pi Han Goh – There were a lot. No, I've not been able to stay in touch with any of them, except Brian Charlesworth, who's active here now, and you.

Michael Mendrin - 5 years, 3 months ago

@Michael Mendrin – Yes, of course there's others as well like kb, Pauley Morph, Morewood, Vašek, husoski, JB, Mathmom, M3, Duke, smci, Zo Maar, Awms A.

Sad, have you made an effort to invite them over? I might consider inviting them over if you haven't done so already.

Pi Han Goh - 5 years, 3 months ago

@Pi Han Goh – I don't even know how to contact them, Pi Han. Remember First Grade Rocks! ? Fred? Quadrillerator? Can't find them either. Oh yeah, let's not forget gopihan. Whatever happened to him?

Michael Mendrin - 5 years, 3 months ago

@Michael Mendrin – Hmm.... let me send them (all) a private message then.

I think you meant "gôhpihán", I think he developed a Dissociative Identity Disorder and I don't know how to contact him. Sad news, he really is the wisest of them all.

Pi Han Goh - 5 years, 3 months ago

@Pi Han Goh – I think Brilliant.org would greatly benefit if more of the old Y!A regulars came here. Especially that gôhpihán.

Michael Mendrin - 5 years, 3 months ago

How many Heronians

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