Kai Wang,

Heptagonal Triangle and Trigonometric Identities,

Forum Geometricorum, 19 (2019) 29-38.

Abstract. We will study the trigonometric identities for heptagonal triangles.

Let a < b < c be the heptagonal triangle’s sides and let R be the circumradius. We will prove the following:

2b^2 – a^2 = √7bR,   2c^2 – b^2= √7cR,   2a^2 – c^2 = −√7aR.

We will also prove the following trigonometric formula:

4 sin 2kπ/7 – tan kπ/7 =  √7for k = 1,2, 4 and -√7 for k = 3,5,6.

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