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The angles of proper spherical triangles are (by convention) less than π, so that < + + < (Todhunter, [1] Art.22,32). In particular, the sum of the angles of a spherical triangle is strictly greater than the sum of the angles of a triangle defined on the Euclidean plane, which is always exactly π radians.
The large triangle that is inscribed in the circle gets subdivided into three smaller triangles, all of which are isosceles because their upper two sides are radii of the circle. Inside each isosceles triangle the pair of base angles are equal to each other, and are half of 180° minus the apex angle at the circle's center.
An excircle or escribed circle [2] of the triangle is a circle lying outside the triangle, tangent to one of its sides, and tangent to the extensions of the other two. Every triangle has three distinct excircles, each tangent to one of the triangle's sides.
The included angle for any two sides of a polygon is the internal angle between those two sides.) If and only if three pairs of corresponding sides of two triangles are all in the same proportion, then the triangles are similar. [b] Two triangles that are congruent have exactly the same size and shape. All pairs of congruent triangles are also ...
If b ≥ c, then β ≥ γ (the larger side corresponds to a larger angle). Since no triangle can have two obtuse angles, γ is an acute angle and the solution γ = arcsin D is unique. If b < c, the angle γ may be acute: γ = arcsin D or obtuse: γ ′ = 180° − γ.
Ptolemy used a circle of diameter 120, and gave chord lengths accurate to two sexagesimal (base sixty) digits after the integer part. [2] The chord function is defined geometrically as shown in the picture. The chord of an angle is the length of the chord between two points on a unit circle separated by that central angle.
The area of a triangle is proportional to the excess of its angle sum over 180°. Two triangles with the same angle sum are equal in area. There is an upper bound for the area of triangles. The composition (product) of two reflections-across-a-great-circle may be considered as a rotation about either of the points of intersection of their axes.
The angles which the circumscribed circle forms with the sides of the triangle coincide with angles at which sides meet each other. The side opposite angle α meets the circle twice: once at each end; in each case at angle α (similarly for the other two angles).