There is a triangle ABC on a euclidean plane. Like every other triangle on the plane, it follows the law of sines, that is, BC/sin(A) = AC/sin(B) = AB/sin(C).

So we know that these three numbers are equal to one another, but most people don't know that they are also equal to the length of a special line segment. What is the significance of this length, and can you prove it?

Nice problem --it serves to further advertise the Law of Sines, which also was central in Bractal's recent problem "Perpendicluar Area." It is amazing how many interesting triangle problems like this exist  that are simple to state and not too hard to solve, but that state results that are not common knowledge. I particularly like right triangle problems myself, such as the one I posed in "Another Man's Floor," which is a right triangle problem for the "lattice point area" of a right triangle similar to "Dotty Right Triangle."

Edited on December 13, 2005, 2:04 pm

Posted by Richard on 2005-12-13 14:03:45