MHB Can You Prove $AB+BC \ge AD+DC$ in a Triangle?

[anemone]

This week's problem was submitted by Ackbach and we truly appreciate his taking the time to propose a quality problem for us to use as our Secondary School/High School POTW.:)Given a triangle $\Delta ABC$, and a point $D$ inside the triangle, prove that $AB+BC \ge AD+DC$. Here's the catch: see if you can prove it within one hour. Please post your honest solving time along with your solution. --------------------
Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!

 

[anemone]

Congratulations to magneto for his correct solution!:)

Solution from magneto:

Spoiler

Extend the line $AD$ to intersect $BC$; name that intersection $K$. Apply the triangle inequality on triangle $DKC$ and $AKB$, we have that $DC \leq DK + KC$ and $AK \leq AB + KB$. Thus,

$AD + DC \leq AD + DK + KC = AK + KC \leq AB + KB + KC = AB + BC$.

We'd like to express our thanks to Ackbach again for his suggested problem.:)