Problem of the Week # 142 - December 15, 2014

[Ackbach]

Here is this week's University POTW:

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I have to admit this is something of an experiment. Please put aside your ego, and bear with me. You MUST record (be honest, now!) the amount of time it takes you to solve this problem. Any solution not including the time spent I will consider incorrect, and I will not publish it. Also, to be clear, I will not publish your times. If I get enough submissions, I will publish a summary of the time statistics. Indeed, if you spend time on this problem, but do not submit a solution, I'd love to have that data as well (again, I will not publish the specifics). In other words, this data I am regarding as anonymous.

Let $ABC$ be an arbitrary triangle, with an interior point $D$. Prove that
$$AB+BC\ge AD+DC.$$

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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!

 

[Ackbach]

This problem is on page 4 of How to Solve It: Modern Heuristics by Michalewicz and Fogel.

No one answered this week' POTW. Here is my solution. Full disclosure: I spent hours and hours on this problem, and didn't even solve it on my own.

Spoiler

Extend line $AD$ to line $BC$, intersecting at $E$ (there's a theorem that let's you do this). Then:
\begin{align*}
AD+DC&\le AD+DE+EC \quad \text{by the Triangle Inequality} \\
&=AE+EC \\
&\le AB+BE+EC \quad \text{by the Triangle Inequality} \\
&=AB+CB.
\end{align*}
So you see, this solution is extremely simple, but when you yank this problem out of a typical context (high school geometry textbook), you make the problem considerably more difficult.