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Van Schooten's and Pompeiu's Theorems: What are these?
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Van Schooten's and Pompeiu's Theorems
The applet illustrates two theorems discovered three hundred years appart. Let P be a point in the plane of an equilateral ΔABC. Both theorems are concerned with the relative lengths of the segments AP, BP, and CP, joining P the vertices of ΔABC.
Theorem (F. van Schooten, 1615 – 1660)
For P on the circumcircle of ΔABC. The longest of the three segments equals the sum of the shorter two.
Theorem (D. Pompeiu, 1873 - 1954)
For P not on the curcumcircle of ΔABC, there exists a triangle with sides PA, PB, and PC.
(The triangle in question is known as Pompeiu's triangle.)
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Ptolemy's theorem offers one approach to proving the theorems. A proof below is more elementary.
Rotate the plane 60o around B so as to make vertex C overlap vertex A. Let A' and P' be the images of A and P, respectively. We'll show that APP' is Pompeiu's triangle.
By the construction, ΔBPP' is equilateral, so that
Assuming A, P, and P' are collinear (i.e., when Pompeiu's triangle is degenerate), 
APB is either 60o or 120o. (This is because it is either equal to BPP', which is 60o, or is supplementary to it.) But then the quadrilateral ABCP is cyclic, proving van Schooten's theorem and its converse.
Remark
van Schooten's theorem admits a non-trivial extension for triangles that are not equilateral!
References
- T. Andreescu, R. Gelca, Mathematical Olympiad Challenges, Birkhäuser, 2004, p. 4.
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