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A note on special cases of Van Aubel’s theorem. (English) Zbl 1477.51010

Van Aubel’s theorem states that the two line segments joining the centers of the squares constructed externally on the opposite sides of a convex quadrilateral \(\mathcal Q\) are perpendicular and have the same length. This paper considers general cases of the theorem. After stating preliminaries, where the last corollary is incorrect, the author considers four equilateral triangles constructed all externally or all internally on the sides of \(\mathcal Q\), and shows that the line joining a pair of the vertices of the triangles, which are not the vertices of \(\mathcal Q\), on the opposite side of \(\mathcal Q\) is perpendicular to the line joining the centers of the remaining two triangles. Then four parallelograms are constructed by \(\mathcal Q\) and the eight equilateral triangles with their centers, where the four points of intersection of the diagonals are collinear. In the next part, the paper considers slightly generalized triangles and their circumcenters instead of the equilateral triangles and their centers, and asserts similar theorems stated in the previous part and existence of several sets of collinear points.
The paper is redundant and the figures are inaccurate. One figure in this paper is consisting of \(\mathcal Q\) with four points which are used to construct the four triangles on the sides of \(\mathcal Q\). However, the author denotes those points in two ways (i) \(P\), \(Q\), \(R\), \(S\) and (ii) \(P\), \(Q\), \(R\), \(T\). Theorem 3.1 is stated using both (i) and (ii) at the same time, i.e., the presentation is wrong. Figures 3 and 4 describe the situations of Theorems 3.4 and 3.5, but the notations are not consistent with those in the theorems, because the two figures use (i) while the theorems use (ii). Theorems 3.1 and 3.2 assume that the side lengths of \(\mathcal Q\) equal \(a\), \(b\), \(c\) and \(d\), but the assumption is not used in the proofs.

MSC:

51M04 Elementary problems in Euclidean geometries
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References:

[1] Dao Thanh Oai, Generalizations of some famous classical Euclidean geometry theorems, International Journal of Computer Discovered Mathematics (IJCDM) 1(3) (2016) 13-20. · Zbl 1347.51005
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[5] Michael de Villiers, Dual generalizations of Van Aubel’s theorem, The Mathematical Gazette November (1998) 405-412.
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[8] http://www.cut-the-knot.org/Curriculum/Geometry/EquiTriOnPara.shtml.
[9] http://dynamicmathematicslearning.com/vanaubel-application.html.
[10] http://dynamicmathematicslearning.com/aubelparm.html.
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