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?

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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?

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For clarification, does it mean that each altitude must be shorter than

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?

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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

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

I'm considering using this as my signature.

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