Main Concept

Morley's trisector theorem claims that in any triangle, the points of intersection of the adjacent triangle trisectors form an equilateral triangle. This theorem was discovered in 1899 by Frank Morley. Many proofs of Morley's theorem have been developed, and while the proofs are often technical, an interesting result that arises from the proof is the following equation for the side length of the equilateral triangle that is formed:

$\mathbf{s}\mathbf{ide}\mathbf{ }\mathbf{length}\mathbf{\=}\mathbf{8}\mathbf{ }\mathit{R}\mathbf{ }\mathbf{sin}\left(\frac{\mathit{A}}{\mathbf{3}}\right)\mathbf{sin}\left(\frac{\mathit{B}}{\mathbf{3}}\right)\mathbf{sin}\left(\frac{\mathit{C}}{\mathbf{3}}\right)$

where A, B, and C are the angles of the larger triangle, and R is the radius of the circumcircle.

Use the sliders to visually verify Morley's theorem for different shapes of triangles.

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