An analogue of the Pythagorean theorem with regular \(n\)-gons instead of squares
A triangle \(ABC\) with angle \(\gamma=2 \pi/n\) at \(C\) is said to be a fraction \(n\)-triangle, and one with \(\gamma= \pi-2 \pi/n\) is called a complementary fraction \(n\)-triangle. Furthermore, the side \(c\) opposite to \(C\) is the hypotenuse of \(ABC\), and the side \(c'= A'B\) of \(A'BC\) (where \(A'\) is obtained by reflecting \(A\) at \(C\), such that \(c'\) satisfies the equality \((c')^2= 2(a^2+ b^2)-c^2)\) is the corresponding complementary hypotenuse. The other two sides are called legs. The author proves the following Pythagorean theorem with regular \(n\)-gons: Let \(ABC\) be a fraction \(n\)-triangle (resp. comlementary fraction \(n\)-triangle). Then the area of a regular \(n\)-gon over the sum of the legs equals the sum of \(n\) areas of the triangle and the area of a regular \(n\)-gon over the complementary hypotenuse (resp. hypotenuse).
scientific article; zbMATH DE number 1036783