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?

(In reply to A (probably wrong) quick guess by Rollercoaster)

Construct the circumscribing circle.  Then construct a perpendicular to segment BC at point B.  Call its intersection with the circle A'. Connect C to A', forming triangle A'BC. A and A' are both on the circumference of the circle, and angles A and A' are equal as they each subtend chord BC.

The sine of A' equals BC / CA', but CA' is a diameter of the circumscribing circle as it is the hypotenuse of a right triangle inscribed in the circle.  That makes this diameter, CA'=BC/sin A'. But since A = A', the diameter of the circumscribing circle is BC/sin A.

Posted by Charlie on 2005-12-13 12:02:42