Proof Without Words: A Sine Identity for Triangles Roger B. Nelsen ([email protected]), Lewis & Clark College, Portland OR If x + y + z = π, then 4(sin x)(sin y)(sin z) = sin 2x + sin 2y + sin 2z. Proof.

1 1 1 1 (a) (2R sin y)(2R sin z) sin x = R 2 sin 2x + R 2 sin 2y + R 2 sin 2z, 2 2 2 2 1 1 2 1 2 1 2 (b) (2R sin y)(2R sin z) sin x = R sin 2y + R sin 2z − R sin(2π − 2x). 2 2 2 2 The identity holds for all real x, y, z such that x + y + z = π. Summary. We wordlessly prove a sine result for the angles of a triangle. http://dx.doi.org/10.4169/college.math.j.45.5.376 MSC: 51M15

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