Viviani by Inversion

Ptolemy's theorem can be proved by inversion from a simple identity AB + BC = AC, where point B is between A and C and all three are collinear. By exactly same reasoning we can prove another non-trivial statement:

Let point M lie on the arc between vertices A₁A_n of the circumcircle of a regular n-gon A₁A₂...A_n. Let d_j denote the distance from M to A_j, j = 1, ..., n. Then

(1)	1 d1d2 + 1 d2d3 + 1 d3d4 + ... + 1 dn-1dn = 1 d1dn

Solution

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Copyright © 1996-2012 Alexander Bogomolny

(1)	1 d1d2 + 1 d2d3 + 1 d3d4 + ... + 1 dn-1dn = 1 d1dn

Solution

Perform inversion f with center M and radius 1. Let f(A_j) = B_j, j = 1, ..., n. Then by the distance formula:

B_jB_j+1 = A_jA_j+1/MA_jMA_j+1 = a/d_jd_j+1,

where a is the side length of the given n-gon.

Points B are all collinear stretching successively from B₁ to B_n, implying that

B₁B₂ + B₂B₃ + ... + B_n-1B_n = B₁B_n.

Substitution now gives the desired identity (1).

Note: For n = 3, we may multiply by the product d₁d₂d₃ to obtain

d₃ + d₁ = d₂,

which is Viviani's Theorem.

References

1. I. M. Yaglom, Geometric Transformations IV, MAA, 2009

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Copyright © 1996-2012 Alexander Bogomolny