Viviani's theorem

The set of Viviani , named after the Italian mathematician Vincenzo Viviani (1622-1703) is a simple statement of equilateral triangles :

If any point is inside an equilateral triangle, the sum of the distances between this point and the sides is constant: ${\ displaystyle P}$

{\ displaystyle s + t + u = h = 3r \,}

The height of the triangle and the incircle radius denote . ${\ displaystyle h}$ ${\ displaystyle r}$

This can be made clear geometrically. The area of the equilateral triangle is as large as the sum of the areas of the triangles marked in color.

For the area of the equilateral triangle ABC applies , where the base and the height should be. ${\ displaystyle {\ frac {gh} {2}}}$ ${\ displaystyle g = {\ overline {AB}} = {\ overline {BC}} = {\ overline {CA}}}$ ${\ displaystyle h \,}$

The sum of the areas of the colored triangles is . ${\ displaystyle {\ frac {gs} {2}} + {\ frac {gt} {2}} + {\ frac {gu} {2}}}$

So:

${\ displaystyle {\ frac {gh} {2}} = {\ frac {gs} {2}} + {\ frac {gt} {2}} + {\ frac {gu} {2}}}$

This follows the claim . ${\ displaystyle h = s + t + u}$

Viviani's theorem can be generalized to equilateral and even equiangular polygons .

literature

Heinrich Hermelink: On the history of the set of the lottery sum in the triangle . In: Sudhoffs Archive for the History of Medicine and the Natural Sciences , Vol. 48, H. 3 (September 1964), pp. 240–247 ( JSTOR 20775106 )
Claudi Alsina, Roger B. Nelsen: Charming Proofs: A Journey Into Elegant Mathematics . MAA 2010, ISBN 978-0-88385-348-1 , p. 96 ( excerpt (Google) )
Ken-Ichiroh Kawasaki, Yoshihiro Yagi, Katsuya Yanagawa: On Viviani's Theorem in Three Dimensions . In: The Mathematical Gazette , Vol. 89, No. 515 (Jul., 2005), pp. 283-287 ( JSTOR 3621243 )
Zhibo Chen, Tian Liang: The Converse of Viviani's Theorem . In: The College Mathematics Journal , Vol. 37, No. 5 (Nov., 2006), pp. 390-391 ( JSTOR 27646392 )
Elias Abboud: Viviani's Theorem and Its Extension . In: The College Mathematics Journal , Vol. 41, No. 3 (May 2010), pp. 203-211 ( JSTOR 10.4169 / 074683410x488683 )
Hans Samelson: Proof without Words: Viviani's Theorem with Vectors . In: Mathematics Magazine , Vol. 76, No. 3 (Jun., 2003), p. 225 ( JSTOR 3219327 )

Web links

Commons : Viviani's theorem - collection of images, videos and audio files

Viviani's theorem on cut-the-knot (English)
Eric W. Weisstein : Viviani's Theorem . In: MathWorld (English).

Individual evidence

↑ Michael de Villiers: Crocodiles and Polygons . In: Mathematics in School , Vol. 34, No. 2, Mar. 2005, pp. 2-4 ( JSTOR 30215779 )

[1] Michael de Villiers: Crocodiles and Polygons . In: Mathematics in School , Vol. 34, No. 2, Mar. 2005, pp. 2-4 ( JSTOR 30215779 )