perplexus.info :: Geometry : The Law of Sines

The Law of Sines (Posted on 2005-12-13)

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?

Submitted by Tristan

Rating: 3.0000 (2 votes)

Solution: (Hide)

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.

Comments: ( You must be logged in to post comments.)

Subject	Author	Date
Puzzle Thoughts	K Sengupta	2023-02-16 07:40:53
Nice Problem	Richard	2005-12-13 14:03:45
Solution	Bractals	2005-12-13 13:00:18
re: quick guess -- proof	Charlie	2005-12-13 12:02:42
A (probably wrong) quick guess	Rollercoaster	2005-12-13 11:39:44

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