Show that there is a hyperbolic triangle with angles $\pi/p$, $\pi/q$, $\pi/r$ for positive integers p,q,r if and only if $1/p+1/q+1/r<1$.and that the one with p,q,r=1,3,7 has the smallest area.
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You can construct the triangle explicitly in one of the standard models of $\Bbb{H}^2$, and then use the fact that area = $\pi - $ sum of angles. (c.f. https://math.stackexchange.com/questions/2864/area-of-a-triangle-propto-pi-alpha-beta-gamma?rq=1) If you want to be more algebraic, you could use presentations of triangle groups in $\operatorname{Isom}(\Bbb{H}^2)$, https://en.wikipedia.org/wiki/Triangle_group – Neal Jul 13 '17 at 18:07
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$2,3,7$, maybe? – Oscar Lanzi Jul 19 '17 at 21:27