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Chapter 5, Problem 467(choose chapter or problem)
The ratio (sin A)/a that shows up in the Law of Sines shows up another way in the geometry of \(\triangle A B C\): It is the reciprocal of the radius of the circumscribed circle.
(a) Let \(\triangle A B C\) be circumscribed as shown in the diagram, and construct diameter \(C A^{\prime}\). Explain why \(\angle A^{\prime} B C\) is a right angle.
(b) Explain why \(\angle A^{\prime}\) and \(\angle A\) are congruent.
(c) If a, b, and c are the sides opposite angles A, B, and C as usual, explain why \(\sin A^{\prime}=a / d\), where d is the diameter of the circle.
(d) Finally, explain why (sin A)/a = 1/d.
(e) Do (sin B)/b and (sin C)/c also equal 1/d? Why?
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