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?
The number happens to be equal to the length of the diameter of the circle that is circumscribed about the triangle.
Proof:
Circumscribe a circle about the triangle ABC. Draw the diameter through point C, and call the other end of the diameter point D. The sine of angle A and the sine of angle BDC must be equal. Also, angle CBD must be a right angle.
sin(A) = sin(BDC) = BC/CD
BC/sin(A) = BC * CD/BC = CD
CD is the diameter of the circle.
Credit goes to my math teacher for this puzzle.
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