Class of triangles

luivga

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Apr 8, 2019
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How can be described in other words the class of triangles, in which each altitude is less than each side?
I tried to solve the problem analytically, using the formulas for the altitudes as functions of the sides, but the result was a system of six algebraic inequalities of degree 4. I couldn't solve this system. Maybe there is a pure geometric approach?
 

Dr.Peterson

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Interesting question!

For clarification, does it mean that each altitude must be shorter than its own corresponding side (the side it is perpendicular to), or than all of the sides?

Also, what is the context of the problem? Did you invent it (so that there may be no simple answer), or is it from a textbook, or a contest, or elsewhere?
 

luivga

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Each altitude must be shorter than all of the sides.
It's funny - I dreamt this problem
 

Dr.Peterson

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Someone who dreams problems like this deserves the chance to figure it out on their own!

I've played with it, focusing on the "corresponding side" version to start with, and found some very interesting things (though not yet a characterization of all such triangles). You might consider finding the locus of point C for fixed AB where an altitude equals a side. I was impressed by that. I'll keep thinking, but I hope you'll be thinking about it too!
 

luivga

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Sure, I will. Thank you!
 

Jomo

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Someone who dreams problems like this deserves the chance to figure it out on their own!
Classic quote.
I'm considering using this as my signature.
 

Dr.Peterson

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Of course, I said that (a) as an excuse for not having anything specific to say yet, because I haven't solved more than a little piece of it, and (b) because I wanted to encourage luivga, who seems to have a mathematical mind, to pursue this interesting problem. I'm hoping others are considering it, too. (I'm mostly experimenting on GeoGebra.)
 
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