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Equilateral Triangles On Sides of a Parallelogram: What Is It About?
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Copyright © 1996-2008 Alexander Bogomolny
Equilateral Triangles On Sides of a Parallelogram
The applet suggests the following theorem [Honsberger, pp. 278-279]:
| On the sides of a parallelogram ABCD, construct similarly oriented equilateral triangles ABX, BCY, CDZ, and DAW. Not accidentally, the quadrilateral XYZW happens to be a parallelogram. On its sides, in turn, construct equilateral triangles XYP, YZQ, ZWR, WXS with the orientation opposite to that of ABX, BCY, CDZ, and DAW. Not surprisingly PQRS is again a parallelogram. Surprisingly, PQRS and ABCD coincide. |
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As on several other occasions (e.g., Three Isosceles Triangles, When a Triangle is Equilateral?, and others), we can make a good use of complex numbers. Points X, Y, Z, W are linear combinations of A, B, C, D with complex coefficients:
| (1) |
X = cA + (1 - c)B, Y = cB + (1 - c)C, Z = cC + (1 - c)D, W = cD + (1 - c)A, |
where
c = (1 + I 3)/2,1 - c = (1 - I 3)/2.
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On the other hand, by construction,
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P = cY + (1 - c)X, Q = cZ + (1 - c)Y, R = cW + (1 - c)Z, S = cX + (1 - c)W. |
In terms of A, B, C, D these can be written as
| (2) |
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However, since ABCD is a parallelogram, A - B = D - C, so that, from the first identity in (2),
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and similarly for the other three vertices.
References
- Honsberger, In Pólya's Footsteps, MAA, 1999
Copyright © 1996-2008 Alexander Bogomolny
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