# Triangle

Hello!

I have a problem with this one:

Points P, Q, R lie respectively on sides BC, CA and AB of ABC triangle. AR=RP=PC and BR=RQ=QC. Prove that AC + BC =2 AB

#### chiro

MHF Helper

Can you draw a picture (or do you have one)?

This is one of the best techniques to solving problems (not just mathematical by the way, but problems in general) and this is the first thing you should do any of these sorts of geometry problems.

My math's teacher said that I should something extra draw to this picture but I don't have any idea. Mayby you know what?

#### chiro

MHF Helper
There is a way you can do a simple proof with vectors, but I will ask if this is OK with you. I understand that in high school, vectors aren't taught that often so if this is a class exercise (and I assume it is), then you will need to use other techniques.

If you can't use the vector approach, you should outline the kinds of techniques you are covering at the moment.

I can use the vectors but how can I do it?

#### chiro

MHF Helper
You have AB + BC = AR + RP + PC and BA + AC = BR + RQ + QC as vectors (but not as lengths). Also BA = -AB since reversing directions reverses the vector.

Now add these together and consider the lengths of the vector (AB + BA + BC + AC = AB - AB + BC + AC = BC + AC).

Ok, I got that vectors AR+RP+PC+BR+RQ+QC=BC+AC, but I still don't got AC+BC=AB. How can I consider the lengths of the vector?