Inequalities in geometry?

Question

Prove : If P is any point inside triangle XYZ, then ZX + ZY > PX + PY. Please do a two collumn proof, or a paragraph proof, but dont miss the main steps and/or assume something that cant be proven.

Everyone in my Honors Geometry class has tried this problem and so far no-one can get it. Can anyone please help me with this?

Answer 1

The proof first involves proving that for any triange xyz, any two sides together is longer than the 3rd, as in xy + yz > zx. Then for the P inside XYZ, construct a parallogram ZAPB, where ZA || PB and ZB || PA. Thus, 2 trianges have been formed, APX and BPY. Then we know that AP + AX > XP and BP + PY > YP, and the rest of the proof follows.

Answer 2

I highly suggest you examine the cosine rule (or more simply, on a right-triangle, the Pythagorean theorem). I will not give the full proof for you, as you should be able to figure it out based upon simple logical abduction.