Orthodiagonal square

An orthodiagonal square. According to the characterization of these quadrilaterals, the two red-colored squares on two opposite sides of the quadrilateral together have the same area as the two blue-colored squares on the other two sides.

In Euclidean geometry , an orthodiagonal quadrangle is a rectangle in which the diagonals cross at right angles. In other words, it is a four-sided planar figure in which the connecting lines between the non-adjacent corners are orthogonal to each other.

Special orthodiagonal squares are dragon squares , especially diamonds and squares .

Characterizations

A square is orthodiagonal if and only if the sums of the squares of the side lengths match for the two pairs of opposite sides. Denoting the square sides, as usual, sequentially with , , and so this condition is: ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle c}$ ${\displaystyle d}$

${\displaystyle a^{2}+c^{2}=b^{2}+d^{2}}$

The diagonals of a convex quadrilateral are perpendicular to each other if the two bimedians (the connecting lines of opposite side centers) are of equal length.

According to another characterization, the diagonals of a convex square ABCD are perpendicular to one another if and only if

${\displaystyle \angle PAB+\angle PBA+\angle PCD+\angle PDC=\pi }$

applies, where P is the intersection of the diagonals. From this equation it follows almost immediately that the diagonals of a convex quadrilateral intersect at right angles precisely when the projections of the diagonal intersection on the sides of the quadrilateral are the corners of a chordal quadrilateral .

A convex quadrilateral is orthodiagonal if and only if its Varignon parallelogram (the corners of which are the side centers) is a rectangle . A related characterization says that a convex quadrilateral is orthodiagonal if and only if the side centers and the base points of the perpendiculars from the side centers to the opposite sides are concyclic, i.e. lie on a circle ( eight-point circle ). The center of this circle coincides with the center of gravity of the square.

Several conditions for ortho diagonal squares relate to the sub-triangles , , and into which the quadrilateral is divided by its diagonals. Denoting by , , and the connection routes of the diagonal intersection with the midpoints of the sides , , and , and , , and the solders of the rectangle sides so a convex quadrilateral is precisely then ortho diagonal, if one of the following statements is true: ${\displaystyle ABP}$ ${\displaystyle BCP}$ ${\displaystyle CDP}$ ${\displaystyle DAP}$ ${\displaystyle m_{1}}$ ${\displaystyle m_{2}}$ ${\displaystyle m_{3}}$ ${\displaystyle m_{4}}$ ${\displaystyle P}$ ${\displaystyle [AB]}$ ${\displaystyle [BC]}$ ${\displaystyle [CD]}$ ${\displaystyle [DA]}$ ${\displaystyle h_{1}}$ ${\displaystyle h_{2}}$ ${\displaystyle h_{3}}$ ${\displaystyle h_{4}}$ ${\displaystyle P}$ ${\displaystyle ABCD}$

• ${\displaystyle m_{1}^{2}+m_{3}^{2}=m_{2}^{2}+m_{4}^{2}}$
• ${\displaystyle {\frac {1}{h_{1}^{2}}}+{\frac {1}{h_{3}^{2}}}={\frac {1}{h_{2}^{2}}}+{\frac {1}{h_{4}^{2}}}}$

properties

The following applies for the area of ​​an orthodiagonal quadrilateral

${\displaystyle A={\frac {1}{2}}ef}$,

where and stand for the lengths of the two diagonals. ${\displaystyle e}$ ${\displaystyle f}$

Web links

• Martin Josefsson: Characterizations of Orthodiagonal Quadrilaterals. (PDF)
• David Fraivert: Properties of a Pascal points circle in a quadrilateral with perpendicular diagonals. (PDF)
• David Fraivert: A Set of Rectangles Inscribed in an Orthodiagonal Quadrilateral and Defined by Pascal-Points Circles.

Individual evidence

1. ↑ uses e.g. B. in: E. Lampe u. a .: Archive of Mathematics and Physics. Third series, 12th vol., Teubner, Leipzig and Berlin 1907, p. 198 ( online ).
2. ↑ Altshiller-Court, N. (2007), College Geometry, Dover Publications. Republication of second edition, 1952, Barnes & Noble, pp. 136-138.
3. ↑ Ismailescu, Dan; Vojdany, Adam (2009), "Class preserving dissections of convex quadrilaterals", Forum Geometricorum 9: 195-211.
4. ^ Josefsson, Martin (2012), "Characterizations of Orthodiagonal Quadrilaterals", Forum Geometricorum 12: 13-25.
5. ↑ Harries, J. "Area of ​​a quadrilateral," Mathematical Gazette 86, July 2002, 310-311.