Janey - Ouch!!
That's why I'm afraid of those things.
Mirithorn - Since you asked......
The question was to compute the average distance between a fixed point P on a unit circle and any point inside of that circle. So instead of just using a double integral to find it like we were supposed to, I thought it would be more interesting (and possibly easier given my integration skills) to use a geometric proof to show that the average distance would be equal to the radius (1), upon which my intuition insisted. So I started out by making a concentric circle of radius r inside of the original circle, and showed that if you draw a straight line from P, through the center, and through the inner circle, you get two points of intersection of the inner circle and the line, and their average distance from P is 1 (the radius of the outer circle). So then all I had to do was show that if you draw another line from P that is tangent to the inner circle, the distance from P to the tangent point on the inner circle is also equal to 1, and if I could do that, then I could show that for every point on the smaller circle, there is another point on it such that when you take the average of their distances from P, you get 1, and that this would hold true for any r. So I spent a substantial amount of time trying to figure out how to prove that the distance from P to the tangent point on the inner circle would be equal to 1, before I realized that
the tangent line to a circle is perpendicular to its radius so by the Pythagorean theorem, this segment would be
smaller than 1, thus blowing my entire proof to smithereens and making me decide to just do the stupid double integral. Stupid Pythagorean theorem.
Head Gunner for the Royal Sloop Crimson Steel, Queen of the Dead, Instigator of Chaos and Confusion, Knight of the Grand Recursive Order of the Lambda Calculus, and also The Non.
Remember Emu's face, people; one day it's going to be on the news alongside a headline about blowing some landmark to smithereens, and then we can all sigh and say, "She was such a normal person".....
....We'd be lying though.
-Laughlyn
If my doctor tells me to exercise, I am going to force him to do my homework.
-Mirithorn
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