Wheels of Powered Triangles

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Problem: Divide a polygon (other than a right triangle) into similar triangles that are not congruent and not right triangles. How can this be done?


Some triangles have sides . In terms of a value , they have power sides . If all sides are multiplied by , one gets the similar triangle . This is called a powered triangle (not to be confused with a power triangle, used for AC circuits). For the given problem, powered triangles can simplify the solution process.

When powered triangles fit perfectly around a vertex, they make a wheel graph. Sometimes these wheels can lead to solutions when more triangles are added. For example, wheel 5 based on with powers and additions gives a solution when is also added. If the original figure is a triangle, it cannot be similar to the component triangles.

Wheel 1 based on with powers and additions can have added to make a triangle, originally found by Andrzej Zak [1].


Contributed by: Ed Pegg Jr (June 2016)
Open content licensed under CC BY-NC-SA




[1] A. Zak, "A Note on Perfect Dissections of an Equilateral Triangle," The Australasian Journal of Combinatorics, 44, 2009 pp. 87–93. https://ajc.maths.uq.edu.au/pdf/44/ajc_v44_p087.pdf.

[2] E. Pegg Jr. "Zak's Triangle" from Wolfram Community—A Wolfram Web Resource. (Jun 2, 2016) community.wolfram.com/groups/-/m/t/851275.

[3] S. Anderson. "Tilings by Triangles." (May 23, 2016) squaring.net/tri/twt.html.

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