相关论文: Brahmagupta quadrilaterals with equal perimeters a…
A quadrilateral is said to be rational if its four sides, the two diagonals and the area are all expressible by rational numbers. The problem of constructing rational quadrilaterals dates back to the seventh century when Brahmagupta gave an…
This note gives a very succinct conceptual proof of Brahmagupta's formula for cyclic quadrilaterals, and then a long discussion about how the proof is inspired by ideas from modern mathematics.
It is well known that Heron's theorem provides an explicit formula for the area of a triangle, as a symmetric function of the lengths of its sides. It has been extended by Brahmagupta to quadrilaterals inscribed in a circle (cyclic…
A triangle with rational sides and rational area is called a rational triangle. In this paper we consider three problems of finding pairs of rational triangles which have a common circumradius as well as either a common perimeter or a…
In this paper we consider the problem of finding pairs of triangles whose sides are perfect squares of integers, and which have a common perimeter and common area. We find two such pairs of triangles, and prove that there exist infinitely…
The diagonals of a quadrilateral form four associated triangles, called half triangles. Each half triangle is bounded by two sides of the quadrilateral and one diagonal. If we locate a triangle center (such as the incenter, centroid,…
Recent analyses of Brahmagupta's discourse on the cyclic quadrilateral, and of Baudh\=ayana's approximate quadrature of the circle, have shown that it is useful to submit mathematical texts to a form of literary analysis. Several passages…
In this paper we obtain several parametric solutions of the quartic diophantine equation $(x_1^4+x_2^4)(y_1^4+y_2^4)=z_1^4+z_2^4$. We also show how infinitely many parametric solutions of this equation may be obtained by using elliptic…
In this paper we obtain cyclic pentagons and hexagons with rational sides, diagonals and area all of which are expressed in terms of rational functions of several arbitrary rational parameters. On suitable scaling, we obtain cyclic…
The diagonals of a quadrilateral form four component triangles (in two ways). For each of various shaped quadrilaterals, we examine 1000 triangle centers located in these four component triangles. Using a computer, we determine when the…
In Euclidean geometry, a bicentric quadrilateral is a convex quadrilateral that has both a circumcircle passing through the four vertices and an incircle having the four sides as tangents. Consider a bicentric quadrilateral with rational…
Four points ordered in the positive order on the unit circle determine the vertices of a quadrilateral, which is considered either as a euclidean or as a hyperbolic quadrilateral depending on whether the lines connecting the vertices are…
The circumcircle of a planar convex polygon P is a circle C that passes through all vertices of P. If such a C exists, then P is said to be cyclic. Fix C to have unit radius. While any two angles of a uniform cyclic triangle are negatively…
A Heron quadrilateral is a cyclic quadrilateral whose area and side lengths are rational. In this work, we establish a correspondence between Heron quadrilaterals and a family of elliptic curves of the form $y^2 = x3+/alpha x^2-n^2x.$ This…
We give an algebraic identity for cubic polynomials which generalizes Brahmagupta's identity and facilitates arithmetic in cubic fields. We also pose a question about a relationship between the elements of a cubic field of fixed trace and…
A convex quadrilateral with sides a,b,c,d, and diagonals p,q is cyclic iff abp-bcq+cdp-daq=0. This condition, in spite of its simplicity, appears to be unnoted and unexpectedly proof-resilient. We employ advanced methods of computer algebra…
Nandakumar asked whether there is a tiling of the plane by pairwise non-congruent triangles of equal area and equal perimeter. Here a weaker result is obtained: there is a tiling of the plane by pairwise non-congruent triangles of equal…
We consider right prisms with horizontal quadrilateral bases and tops, and vertical rectangular sides. We look for examples where all the edges, face diagonals and space diagonals are integers. We find examples when the base is an isosceles…
A certain real number, depending on two neighbouring sides of a quadrilateral and the diagonal meeting these two sides at their common point, is shown to be invariant under affinity. As an application we demonstrate a nice formula for the…
Generic spherical quadrilaterals are classified up to isometry. Condition of genericity consists in the requirement that the images of the sides under the developing map belong to four distinct circles which have no triple intersections.…