sinh (a/γ) / sin A = sinh (b/γ) / sin B = sinh (c/γ) / sin C,
(4)
and also
cosh (a/γ) = cosh (b/γ) cosh (c/γ) − sinh (b/γ) sinh (c/γ) cos A,
(5)
| Fig. 68. |
with two similar equations. The sum of the three angles of a triangle is always less than two right angles. The area of the triangle ABC is λ²(π − A − B − C). If the base BC of a triangle is kept fixed and the vertex A moves in the fixed plane ABC so that the area ABC is constant, then the locus of A is a line of equal distance from BC. This locus is not a straight line. The whole theory of similarity is inapplicable; two triangles are either congruent, or their angles are not equal two by two. Thus the elements of a triangle are determined when its three angles are given. By keeping A and B and the line BC fixed, but by making C move off to infinity along BC, the lines BC and AC become parallel, and the sides a and b become infinite. Hence from equation (5) above, it follows that two parallel lines (cf. Section VII. Axioms of Geometry) must be considered as making a zero angle with each other. Also if B be a right angle, from the equation (5), remembering that, in the limit,
cosh (a/γ) / cosh (b/γ) = cosh (a/γ) / sinh (b/γ) = 1,
we have