geometry - prove that you can make new triangle with..

Question

I have a triangle, the length of heights are $i,h,g$.

Prove that we can build a new triangle so that the lengths of the sides are: $i^{-1}, g^{-1}, h^{-1}$ (see picture)

Answer 1

Let the respective bases for the heights $\displaystyle h,g,i$ be $\displaystyle H, G, I$, and the latter three are the sides of the original triangle.

By considering the area of the triangle, establish that $\displaystyle hH = gG = iI$.

Rearrange (for example) to $\displaystyle \frac{1}{h} = (\frac{H}{G})\frac{1}{g}$ and similarly show that $\displaystyle \frac{1}{i} = (\frac{I}{G})\frac{1}{g}$

Now let $\displaystyle \frac 1h + \frac 1i - \frac 1g = \alpha$

This gives $\displaystyle H + I - G = \alpha Gg$ and by using the triangle inequality on the original triangle, show that $\displaystyle \alpha > 0$.

Proceed similarly for the other sides (or conclude by symmetry). You have now established the triangle inequalities for the new triangle, and you're done.

Answer 2

Each side of the original teiangle is twice the area divided by the altitude to that side. So your proposed triangle is similar to the original one.