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  "body": "\n## Definition\n\nLet $\\mathbb{H}^2$ be the [[hyperbolic plane|GN4N94GQ0Z]] of constant curvature $-1$.\nLet $A$, $B$, $C$ be three points of $\\mathbb{H}^2$ that do not lie on one [[geodesic|NSYMQMHPDN]] line.\nThe *hyperbolic triangle* $ABC$ is the union of the three geodesic segments $AB$, $BC$ and $CA$, together with the region they bound.\nIts sides are the hyperbolic lengths $a = d(B, C)$, $b = d(C, A)$, $c = d(A, B)$.\nIts angles $\\alpha$, $\\beta$, $\\gamma$ at $A$, $B$, $C$ are the angles between the tangent vectors of the two sides at each vertex.\n\n## Remarks\n\nExample: in the [[Poincaré disk model|85MM2EA0TK]], take $A = 0$, $B = 1/2$ and $C = i/2$.\nThe only geodesic through $0$ and $1/2$ is the real diameter, which does not contain $i/2$, so $A, B, C$ span a hyperbolic triangle.\nIts angle at $A$ is $\\pi/2$, since the model is conformal and the sides at $A$ are perpendicular diameters.\nNon-example: the points $0$, $1/4$, $1/2$ lie on the real diameter, which is one geodesic line, so they do not form a hyperbolic triangle.\nIn curvature $-1$, the angles satisfy $\\alpha + \\beta + \\gamma \u003c \\pi$, and the area is $\\pi - (\\alpha + \\beta + \\gamma)$.\nOne also allows vertices at infinity, on the ideal boundary; a triangle with all three vertices ideal is an *ideal triangle*, with all angles $0$ and area $\\pi$.\nThe notion is the hyperbolic counterpart of the Euclidean [[triangle|0CTNFGYMM2]].\n",
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