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Equivalently, a quadrilateral has equal diagonals if and only if it has perpendicular bimedians, and it has perpendicular diagonals if and only if it has equal bimedians. [7] Silvester (2006) gives further connections between equidiagonal and orthodiagonal quadrilaterals, via a generalization of van Aubel's theorem. [8]
Every kite is an orthodiagonal quadrilateral, meaning that its two diagonals are at right angles to each other. Moreover, one of the two diagonals (the symmetry axis) is the perpendicular bisector of the other, and is also the angle bisector of the two angles it meets. [1] Because of its symmetry, the other two angles of the kite must be equal.
An equilic quadrilateral has two opposite equal sides ... difference of the areas of the two triangles it is composed of. ... if the diagonals are perpendicular and ...
Using congruent triangles, one can prove that the rhombus is symmetric across each of these diagonals. It follows that any rhombus has the following properties: Opposite angles of a rhombus have equal measure. The two diagonals of a rhombus are perpendicular; that is, a rhombus is an orthodiagonal quadrilateral. Its diagonals bisect opposite ...
This follows from the Pythagorean theorem, by which either of these two sums of two squares can be expanded to equal the sum of the four squared distances from the quadrilateral's vertices to the point where the diagonals intersect. Conversely, any quadrilateral in which a 2 + c 2 = b 2 + d 2 must be orthodiagonal. [4]
This is a list of two-dimensional geometric shapes in Euclidean and other geometries. For mathematical objects in more dimensions, see list of mathematical shapes. For a broader scope, see list of shapes.
The areas S and T of some two opposite triangles of the four triangles formed by the diagonals satisfy the equation = +, where K is the area of the quadrilateral. [16]: Thm.8 The midpoints of two opposite sides of the trapezoid and the intersection of the diagonals are collinear. [16]: Thm.15
The sum of the squared lengths of any two perpendicular chords intersecting at a given point is the same as that of any other two perpendicular chords intersecting at the same point, and is given by 8r 2 – 4p 2 (where r is the circle's radius and p is the distance from the center point to the point of intersection). [5]