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Let a,b,c denote the lengths of the sides and let A,B,C denote the corresponding opposite angles of a triangle. In the Euclidean plane we have the law of sines. a/sin(A)=b/sin(B)=c/sin(C). A recent article pointed out that these 3 equal ratios are also all equal to the diameter of a circle whose circumference contains the 3 vertices of the triangle. There is a similar law of sines for spherical triangles. sin(a)/sin(A)=sin(b)/sin(B)=sin(c)/sin(C). Are these 3 equal ratios also all equal to some important geometric quantity associated with the spherical triangle? If so, what is this quantity and does the same sort of phenomenon (involving an analogous "law of sines") occur with triangles in some other metric spaces-such as hyperbolic space?

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up vote 7 down vote accepted

I am amazed with your statement that "a recent article pointed...". I suspect his fact was known to the Greeks about 2000 years ago, and it is certainly mentioned in every serious trigonometry textbook.

On your question. The corresponding quantity for the spherical triangle even has a name: "modulus of the triangle", M. There is a symmetric expression $$M^2=\frac{1-\cos^2a-\cos^2b-\cos^2c+2\cos a\cos b\cos c}{\sin^2a\sin^2b\sin^2c},$$ where $a,b,c$ are the sides.

Modulus plays a role in various theorems. For example, If the modulus is greater than $1$, then either all 3 sides are greater than 90 degrees, or exactly one of the sides is less than 90 degrees.

Modulus is equal to the ratio of absolute values of two determinants: $M=|\delta/\Delta|$ where $\delta$ is the determinant of the three unit vectors pointing from the center of the unit sphere to the vertices, and $\Delta$ is the determinant of the three unit vectors which are pointing to the vertices of the polar (dual) triangle.

For this fact, see M. Berger, Geometrie, vol. 2. (The relatively modern book covering the subject). The most comprehensive book in English is W. Cahuvenet, A treatease on Plane and Spherical trigonometry, Philadephia 1850. (Poincare once said that this book contains everything one may want to know on the subject:-)

EDIT. Just found more information on this quantity. Will just give a reference: Study, Spharische Trigonometrie, orthogonale Substitutionen und elliptische Funktionen, Leipzig, 1893.

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I really appreciate this information. I was getting nowhere trying to figure out some geometric meaning for the spherical law of sines. The article (actually a book review) that started me on this is on page 707 of the "American Mathematical Monthly" for October 2012. The author felt that while most mathematicians carried the plane law of sines around in their heads-so to speak, the majority would not remember or be aware of the geometrical significance of these ratios. This was certainly true in my case. – Garabed Gulbenkian Oct 19 '12 at 15:15

This is a duplicate of this question, which has good answers.

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Many thanks to you also. I did not realize that there was so much information around on this topic. – Garabed Gulbenkian Oct 19 '12 at 15:22

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