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Regular polygons on sides of special triangles

Published online by Cambridge University Press:  01 August 2016

Zvonko Čerin
Zvonko Čerin*
Affiliation:
Kopernikova 7, 10010 Zagreb, Croatia, e-mail: CERIN@MATH.HR
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Extract

For a natural number n ≥ 3 and a positive real number ξ let Pn (ξ) denote the area of the regular n-gon with side of length ξ and let P(ABC … Z) be the area of a convex polygon ABC … Z.

In 1996 Josip Kovačević observed [1] that in every triangle ABC with sides a, b and c and with the angle A equal to π/3 radians the following relation holds:

Similarly, if the angle A is equal to 2π/3 radians then

On the other hand, when the angle A is π/2 radians, then Pythagoras’ theorem implies which could be rewritten as so

because P(ABC) = bc/2. The last observation is the starting point for Veljan’s results in [2] and [3] that

when the angle A is π - 2π/n radians and that

when the angle A is 2π/n radians.

Type
Research Article
Information
The Mathematical Gazette , Volume 84 , Issue 500 , July 2000 , pp. 260 - 264
Copyright
Copyright © The Mathematical Association 2000

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References

1. Kovaöevic, Josip and Veljan, Darko, A triangular analogue of the Pythagorean theorem, Math. Gaz. 80 (November 1996), pp. 550–554.Google Scholar
2. Veljan, Darko, An analogue of the Pythagorean theorem with regular n-gons instead of squares, Elemente der Mathematik 51 (1996), pp. 156–160.Google Scholar
3. Veljan, Darko, Poopceni Pitagorin pouòak s pravilnim poligonima umjesto kvadratima, Matematiäko-fiziäki list 46 (1995/96), pp. 134–139.Google Scholar