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Hint to puzzle 136: Point in a triangle

Let anglePBC = x°, so that angleBCP = 54° − x°.  Apply the law of sines (also known as the sine rule) to triangles PAB, PBC, and PCA, and hence show that sin 27° · sin(54° − x°) · sin 18° = sin 24° · sin x° · sin 57°.  Then show that x = 15° satisfies this equation.