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Suppose we have a triangle, call it triangle $XYZ$, and a point $W$ inside triangle $XYZ$. How would I prove that $XY + YZ > XW + WZ$? So the way I labeled everything, point $X$ is the bottom left corner, point $Y$ is the top point, and point $Z$ is the bottom right corner where the triangle is sitting flat.

I tried numerous variations of triangle inequality but could not get the result. I am wandering if I need to use something else to prove it.

Nick Freeman
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  • this doesn't help at all, WY has positive length, and that is creating the problem, becasue in the way you proposed, we would get $XY+YZ+2WY>XW+WZ$ – QED Aug 28 '14 at 21:23
  • similar/same problem http://prntscr.com/46y3tc – AgentS Aug 28 '14 at 21:26

1 Answers1

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Let $U$ be the intersection point of the line $XW$ and the edge $YZ$.

enter image description here

Now, for $\triangle XYU$, we have $$XY+YU\gt XW+WU\tag1$$

For $\triangle WUZ$, we have $$WU+UZ\gt WZ\tag2$$

Calculating $(1)+(2)$ gives us $$XY+YU+UZ\gt XW+WZ\iff XY+YZ\gt XW+WZ.$$

P.S. I used $AB+AC\gt BC$ for a $\triangle ABC$.

mathlove
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