Does every hyperbolic triangle have an inscribed circle? Circumscribed circle?

Expert Solution

Each hyperbolic triangle has an inscribed circle but not every hyperbolic triangle has a circumscribed circle .

we can always draw a triangle inside of a hyperbolic triangle as in hyperbolic geometry number of sides must grow with radius of the circle. The reason is the divergence of geodesics from one another. When trying to circumscribe an traingle , we pick 3 equally spaced points on the circle and draw geodesics tangent to the circle at those points. The issue is that when radius is large enough, those geodesics never meet each other and no polygon is formed. so we take radius as small as possible.and we can draw circle inside a traingle.

The hyperbolic triangle ΔABC has a hyperbolic circumcircle if and only if
H(s(AB),s(BC),s(CA)) > 0.
If the condition is satisfied, then the hyperbolic radius of the circumcircle
is r, where sinh²(r) = 4s²(AB)s²(BC)s²(CA)/H(s(AB),s(BC),s(CA)).

Note that the expression for sinh²(r) is equal to (2R)², where R is the
radius of the euclidean circumcircle of the triangle with sides of length
s(AB),s(BC),s(CA). This triangle exists as H(s(AB),s(BC),s(CA)) > 0

so if the condition above is not satisfied by a triangle then it does not have a circumscribed circle

Colby Messinger answered 2 years ago

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