1. ## Triangle inequality

Please, guys, I would aprecciate any help with this problem:
Prove that the sum of the distances of any point within a triangle, to its vertices, is smaller than the triangle perimeter. I know that I have to use the triangle inequality, but I am having a hard time trying to prove the result (which seems quite obvious).
Thanks a lot.

2. Ok, here's what to do.

First, draw up a triangle, like this one:

After viewing the picture, we can show that $\displaystyle b+c>y+z$ because the point where y and z meet is closer to the side a than the point where b and c meet.

In a similar fashion, we can show that
$\displaystyle a+c>x+y$
$\displaystyle a+b>x+z$

After that, just add the three inequalities together, and simplify.

3. Hello scounged! Thanks for the reply.
I know that it is quite obvious that b + c > y + z , but is there a more formal way to prove it?
Thanks again.

4. What do you mean by "more formal way"?

5. Originally Posted by scounged
What do you mean by "more formal way"?
How do you prove the following. No diagrams allowed.
If P is any point interior to $\displaystyle \triangle ABC$, letting
$\displaystyle d_X$ be the distance from P to vertex X that $\displaystyle b+c>d_C+d_B~?$
That is what you claimed. You said the we could see that.
But how do you prove that? No diagrams allowed.

6. Yes, Plato! that's exactly what I meant by "more formal way". I agree with you, scounged, I can see, by looking the picture, that b + c > y + z , but I am afraid that only telling "because the point where y and z meet is closer to the side a than the point where b and c meet" wouldn't be a proof (maybe I'm wrong). Is there any theorem for that?

7. What scounged said is Euclid I.21, which should be easy to find online.. I don't know if you can consider any proof of it "rigorous" according to modern standards without going through quite a bit of preliminary effort first. I know in Hartshorne's Geometry:Euclid and Beyond, this is done by using Hilbert's axioms and maybe a couple more. Whether this is absolutely necessary for this problem, I have no idea.

In any case, whether scounged's suggestion is acceptable or not really depends on the class and what theorems you've proved so far.

8. I think that there's an easy way to prove that $\displaystyle b+c>d_C+d_B$, using the law of cosines. But I don't know. Is there a theorem that states something along these lines?

$\displaystyle \mbox{If}~\mid{a}\mid^x+\mid{b}\mid^x>\mid{c}\mid^ x+\mid{d}\mid^x~\mbox{then}~\mid{a}\mid^{x+n}+\mid {b}\mid^{x+n}>\mid{c}\mid^{x+n}+\mid{d}\mid^{x+n}$

9. Thank you guys so much for the help. I finally found a proof for the Euclid I.21, which directly leads to the proof that the sum of the distances of a point inside a triangle to the vertices is less than the perimeter! I was having nightmares with this problem for days. Here it is: draw the triangle ABC. Let D be a point inside de triangle. Draw BD through toE (where it meats side AC). In triangle ABE, AB + AE > BE . Add EC to each side: AB + AE + EC > BE + EC , so AB + AC > BE + EC (1) . In triangle CED: EC + ED > DC . Add DB to each side: EC + BE > DC + DB (2) . Now, (1) and (2) give us AB + AC > BE + EC > DC + DB . So AB + AC > DC + DB . If we do this with the other sides of the triangle we can show that AD + DB + DC < AB + BC + AC .