Related papers: Revisiting the quadrisection problem of Jacob Bern…

The Miquel-Steiner theorem for a quadrilateral in the Euclidean plane states that the circumcircles of the four component triangles intersect at a single point, which now is called the Miquel-Steiner point of the quadrilateral. In elliptic…

We prove that a surface in real 3-space containing a line and a circle through each point is a quadric. We also give some particular results on the classification of surfaces containing several circles through each point.

Euler's partition identity states that the number of partitions of $n$ into odd parts is equal to the number of partitions of $n$ into distinct parts. Strikingly, Straub proved in 2016 that this identity also holds when counting partitions…

The paper is devoted to the description of family of scalene triangles for which the triangle formed by the intersection points of bisectors with opposite sides is isosceles. We call them Sharygin triangles. It turns out that they are…

We study equivalence relation of the set of triangles generated by similarity and operation on a triangle to get a new one by joining division points of three edges with the same ratio. Using the moduli space of similarity classes of…

We review and comment on some works of Euler and his followers on spherical geometry. We start by presenting some memoirs of Euler on spherical trigonometry. We comment on Euler's use of the methods of the calculus of variations in…

Consider a geodesic triangle on a surface of constant curvature and subdivide it recursively into 4 triangles by joining the midpoints of its edges. We show the existence of a uniform $\delta>0$ such that, at any step of the subdivision,…

In this paper we give a classification of tilings of the sphere by congruent quadrilaterals with exactly two equal edges. The tilings are the earth map tilings, $(p,q)$-earth map tilings and their flip modifications, and quadrilateral…

Given a trapezoid dissected into triangles, the area of any triangle determined by either diagonal of the trapezoid is integral over the ring generated by the areas of the triangles in the dissection. Given a parallelogram dissected into…

The notion of a layered triangulation of a lens space was defined by Jaco and Rubinstein in earlier work, and, unless the lens space is L(3,1), a layered triangulation with the minimal number of tetrahedra was shown to be unique and termed…

A cyclic quadrilateral is called a Brahmagupta quadrilateral if its four sides, the two diagonals and the area are all given by integers. In this paper we consider the hitherto unsolved problem of finding two Brahmagupta quadrilaterals with…

Suppose that $I$ is a unit square. Let $T$ (resp. $\Delta$) be an isosceles right triangle (resp. an equilateral triangle). We prove that any collection of triangles homothetic to $T$ (resp. $\Delta$), whose total area does not exceed…

Given a set of objects $O$ in the plane, the corresponding intersection graph is defined as follows. Each object defines a vertex and an edge joins two vertices whenever the corresponding objects intersect. We study here the case of unit…

Triangles with integer length sides and integer area are known as Heron triangles. Taking rescaling freedom into account, one can apply the same name when all sides and the area are rational numbers. A perfect triangle is a Heron triangle…

In this paper we first study the isoperimetric problem in the case of integer triangles, as well as Alcuin's sequence and how it relates to the number of different integer triangles with a given perimeter. We then present and compare two…

In the 1770s, Euler wrote a series of papers (E563, E691 and E692) about finding the ellipse with minimal area or perimeter in the family of all ellipses passing through a fixed set of points. This is a translation of all three papers from…

The enumeration of the regions formed when circle is divided by secants drawn from points on the circle is one of the examples where the inductive reasoning fails as was pointed out by Leo Moser in the Mathematical Miscellany in 1949. The…

Starting with any nondegenerate triangle we can use a well defined interior point of the triangle to subdivide it into six smaller triangles. We can repeat this process with each new triangle, and continue doing so over and over. We show…

A cubic surface in $P^3$ is known to contain 27 lines, out of which one can form 36 Schlafli double - sixes i.e., collections $l_1,...,l_6, l'_1,..., l'_6\}$ of 12 lines such that each $l_i$ meets only $l'_j, j\neq i$ and does not meet…

We completely classify edge-to-edge tilings of the sphere by congruent quadrilaterals. As part of the classification, we also present a modern version of the classification of edge-to-edge tilings of the sphere by congruent triangles.…