Law of cosines . The power of the point with respect to the circle is equal to both and . Therefore, which is the law of cosines. Using algebraic measures for line segments (allowing negative numbers as lengths of segments) the case of obtuse angle () and acute angle () can be treated simultaneously. By using the law of sines and knowing that the angles of a triangle must sum to 180 degrees, we have the following system of equations (the three unknowns are the angles): Then, by using the third equation of the system, we obtain a system of two equations in
Hyperbolic triangle side with the corresponding letter) is strictly less than a straight angle. The difference between the measure of a straight angle and the sum of the measures of a triangle's angles is called the defect of the triangle. The area of a hyperbolic triangle is equal to its defect multiplied by the square of : This theorem, first proven by Johann Heinrich Lambert, is related to Girard's theorem in spherical geometry. In all the formulas stated below the sides , , and must be measured in absolute length, a unit so that the Gaussian curvature of the plane is −1.