Van Schooten's and Pompeiu's Theorems
What are these?
A Mathematical Droodle
Explanation
|Activities|
|Contact|
|Front page|
|Contents|
|Geometry|
|Store|
Copyright © 1996-2012 Alexander Bogomolny
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.)
Ptolemy's theorem offers one approach to proving the theorems. A proof below is more elementary.
Rotate the plane 60° 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 PP' = BP. In addition, AP' is the image under the above rotation of CP, so that AP' = CP. The remaining side in ΔAPP' is just AP, and we are done with Pompeiu's theorem.
Assuming A, P, and P' are collinear (i.e., when Pompeiu's triangle is degenerate), ∠APB is either 60° or 120°. (This is because it is either equal to ∠BPP', which is 60°, 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!
(There is another proof.)
References
- T. Andreescu, R. Gelca, Mathematical Olympiad Challenges, Birkhäuser, 2004, p. 4.
Ptolemy's Theorem
- Ptolemy's Theorem
- Sine, Cosine, and Ptolemy's Theorem
- Useful Identities Among Complex Numbers
- Ptolemy on Hinges
- Thébault's Problem III
- Van Schooten's and Pompeiu's Theorems
- Ptolemy by Inversion
- Brahmagupta-Mahavira Identities
- Casey's Theorem
- Three Points Casey's Theorem
- Ptolemy via Cross-Ratio
|Activities|
|Contact|
|Front page|
|Contents|
|Geometry|
|Store|
Copyright © 1996-2012 Alexander Bogomolny
| 
