### Post #84: Bisectors and isoceles triangles

Wednesday, March 24th, 2010Dear Alice,

I was recently plagued with a horrendous disease. It came from my girlfriend’s father. She told him that I liked puzzles, and so he inflicted one upon me.

The puzzle is stated as follows, “Given that two bisectors of a triangle are of equal length, prove that this triangle is an isoceles triangle.” As intuitive as this conclusion is, this turns out not to be so trivial. Such is life, eh? He promised me $10 were I to solve it.

My friends and I joked that this was some sort of test to see if I were worthy enough to date his daughter. I think I got lost somewhere along the way and couldn’t separate the joke from reality. I have a tendency to do things just in case, and this little joke became no exception. I refused to look up the problem online.

I did a bit of thinking, and pretty soon, came to a formulae which I thought was promising. I announced to my girlfriend that I was close. It was only a matter of time before I differentiated the equation to its inevitable conclusion. She passed the message on to her father, who responded with, “OK. David is getting “slightly warm”.” Slightly, eh?

I went to him with my six pages of scrapwork. He looked through it, and was pleased. He liked to see other people’s approaches to the problem. He then told me that he has never solved the problem himself. He’s suffered with it for about twenty years, and just had to give up. A friend of his once showed him a solution by drawing a few circles and making a case with words… but never has anyone presented him with an algebraic solution.

Some days, I feel so close to the answer that I can taste it. I’d be alseep at the wheel (at work) when a new approach would spring to mind.

“Try proving that if it’s not an isoceles, then the bisectors are not equal!”

“Try proving that if it’s an isoceles, then the angles are not divided in half!”

I’ve two diagrams to demonstrate the essence of the construction, and even so, I can’t produce the equivalent algebraic solution which would scream out, “Q.E.D.!”

After a few months of fighting with the riddle, it started to take over my mind. Nevermind sleep, I would get bored with people’s conversation really quickly, and I would find myself daydreaming about the puzzle. People started to notice and I had to explain my absent-mindedness. Why am I so distracted? Well, let me give you a little puzzle. Soon, I had half the office working on it, busy sketching triangles in the margins of their books, insisting that they’ll pull the sword from the stone.

I went to see my parents, who are only in town for a brief stint, and soon enough, we were arguing over the possible approaches to the question. My brother was bothered by how worked up we were getting. “What if you’re just trying to prove something as trivial as 2 + 2 = 4?”

I then proved to him, in no uncertain terms, that indeed 2 + 2 = 4, before returning to my puzzle.

Two weeks ago, a co-worker came to me and said that he looked up the answer and was relieved that he did. The solution isn’t trivial, and isn’t worth fighting for. I gave in. He showed me the graph which visually demonstated the non-trivial solution. The graph was horrendous. Not only were there an infinite set of answers for every angle… the graph looked like a quilt, with little infinities drawn in between the diamond patches!

…Well, you would think that I’d have quit at that point, but no. I swore that all still wasn’t lost! I have three approaches to the problem! I have several restrictions to the variables! I have a fucking cheat sheet filled with all of the different inequalities that I could derive! I can still almost *taste* it! Argh!!!

It was almost midnight. I hadn’t yet packed for San Francisco. Yesterday and today, I came straight home from work, doped up to my eyeballs with caffeine, and continued my work on the problem. It had to stop.

I looked up the problem. I did my best to internalize the solution, but I have yet to reproduce it. Someday I will, but not today. Not tonight. Maybe not even this year, but definitely not until I get back from San Francisco.

Tonight, I say “farewell” to this puzzle, as it goes from malignant to benign until further notice.

David =B~)