Proving Altitude Sum Inequality in a Triangle

[nolachrymose]

Altitude Sum Proof

Hi all,

I have this problem that I have no idea where to start. It asks to prove for a triangle with altitudes h_a, h_b, and h_c, that

$$\frac{1}{h_a} < \frac{1}{h_b} + \frac{1}{h_c}$$

Any idea how to begin this proof? I've tried all sorts of algebra, and utitlizing the Triangle Inequality, but I can't seem to reach this conclusion. Any help is greatly appreciated -- thank you! :)

 

[The Bob]

Hi all,

I have this problem that I have no idea where to start. It asks to prove for a triangle with altitudes h_a, h_b, and h_c, that

$$\frac{1}{h_a} < \frac{1}{h_b} + \frac{1}{h_c}$$

Any idea how to begin this proof? I've tried all sorts of algebra, and utitlizing the Triangle Inequality, but I can't seem to reach this conclusion. Any help is greatly appreciated -- thank you! :)

What do you mean by altitudes $$h_a h_b$$ and $$h_c$$?

The Bob (2004 ©)

 

[robphy]

possible hint: "area"

 

[arildno]

I'll upgrade that into a probable hint

 

[The Bob]

I'll upgrade that into a probable hint

So the h_a, h_b and h_c are all angles?

The Bob (2004 ©)

 

[arildno]

No, they are heights:
Let A be the area.
Then a=A/h_a, b=A/h_b,c=A/h_c
Using the triangle inequality for a,b,c yields the proposition.

 

[nolachrymose]

Sorry I didn't post sooner -- I figured it out on my own a little after I had posted, but didn't have time to post my solution. I used the method Arildno suggested.
Thank you for your input, though! :)