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Decided by exhaustion — page 15

Questions with finitely many cases, settled by going through all of them — and what changes when a claim about every argument becomes a count.

Langley's adventitious angle. The isosceles triangle with apex 20° and base angles 80°, the cevian from B at 60° to the base meeting AC at D, the cevian from C at 50° meeting AB at E, and the angle BDE, computed as 30° to nine places.

Geometry

Thirty degrees without a circle

In a triangle with angles 20°, 80° and 80°, draw cevians at 60° and 50° to the base and the angle between them and the segment they cut off is exactly 30°. Whole-number answers in angle problems usually come from a circle, through the inscribed-angle theorem; this one does not, since the four points are not concyclic. Testing every whole-degree quadrilateral finds 1,002,733 with a whole-degree answer: 939,929 explained by a circle, and 62,804 that are not.

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