9827

Trisecting an Angle Using a Lemniscate

The angle to trisect, say α, is between the longer leg of the carpenter's square (the brown "L" shape) and the polar axis . Translate the carpenter's square so that it touches the curve at the point . The angle between the radius vector and the polar axis is one-third of the given angle .
For any curve in polar coordinates, the tangent of the angle between the tangent and radial line (the angle between vectors and ) at the point is . The lemniscate has the polar equation , and the derivative with respect to is , so . So . Since is obtuse, .
So , and the angle between the tangent and the normal to the radius vector is . But this angle is equal to the angle between the larger leg of the carpenter's square and the radius vector , because these angles have orthogonal legs. So , .

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