Geometry

>>8711526
Bump

>>8711526
what does "larger" mean? how do you compare a triangle and a line?

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10KB, 225x224px

>>8712040
Pretty sure OP means one of the three portions of the circle not a part of the triangle's area.

[eqn]
A > \alpha \\
\alpha = \frac{\pi r^2 - A}{3} \\
A = f(r) = \frac{1}{2}*base*height = \frac{1}{2}[(o+r)(2a)] \\
A = r^2(1+sin \theta)(cos \theta) = r^2(1+\frac{1}{2})(\frac{\sqrt{3}}{2}) = \frac{r^2 \sqrt{27}}{4} \\
\frac{r^2\sqrt{27}}{4} > \frac{\pi r^{2} - \frac{r^2 \sqrt{27}}{4}}{3} \\
3\sqrt{27} > 4\pi - \sqrt{27} \\
~15.6 > ~7.4
[/eqn]

Equilateral triangle case. I'm lazy, sue me.

>>8712040
Larger = more area

Segment = the portion of a circle cut off by a chord

>>8712218
Yes, that's trivial. The point is to prove it for all triangles. Here I will help you.

The problem is equivalent to finding a solution to the inequality

x > 2sin(x)+sin(y)-sin(x+y)

where 0 < x < 2pi/3 and x < y < pi-x/2