Related papers: Other trigonometric proofs of Pythagoras theorem
We give three new proofs of the triangle inequality in Euclidean Geometry. There seems to be only one known proof at the moment. It is due to properties of triangles, but our proofs are due to circles or ellipses. We aim to prove the…
In this article we prove some theorems related to the triplets of triangles, homological two by two. These theorems will be used later to build triplets of triangles two by two tri-homological.
In this article we obtain, using an expression of the digamma function $\psi(x)$ due to Mikolas, integral representations of the zeta function of odd arguments $\zeta(2p+1)$ for any positive value of $p$. The integrand consists of the…
We reconsider Archimedes' evaluations of several square roots in 'Measurement of a Circle'. We show that several methods proposed over the last century or so for his evaluations fail one or more criteria of plausibility. We also provide…
John Conway's Circle Theorem is a gem of plane geometry. The six points formed by continuing the sides of a triangle beyond every vertex by the length of its opposite side, are concyclic. The theorem has attracted several proofs. We present…
We establish a Pythagorean theorem for the absolute values of the blocks of a partitioned matrix. This leads to a series of remarkable operator inequalities.
A novel approach to an old symmetry problem is developed. A new proof is given for the following symmetry problem, studied earlier.
Newton's quadrilateral theorem can be phrased as follows. If H is a circle that is tangent to the four extended sides of a non-parallelogram quadrilateral Q, the center of H lies on the Newton line of Q. We prove that the theorem remains…
Few among us would know that the first mention of the sine and the enumeration of the first sine table are to be credited to Aryabhata. The method to generate this relies on the sine difference formula which is derived using ingenious…
Segre's theorem on ovals in projective spaces is an ingenious result from the mid-twentieth century which requires surprisingly little background to prove. This note, suitable for undergraduates with experience of linear and abstract…
In a previous article, we discussed a paradox in Timaeus' cosmology: that there is no void inside the universe, even though it is entirely filled with polyhedra-a mathematical impossibility (Brisson-Ofman 2025). In the present article, we…
The aim of this note is to show that any algebraic relation over $\overline{\mathbb{Q}}$ between the values of the trigonometric functions sine and cosine at algebraic points can be derived from the Pythagorean identity and the angle…
In this expository article we provide an elegant proof of the one-sided Ingham-Karamata Tauberian theorem. As an application, we present a short deduction of the prime number theorem.
The first and second representation theorems for sign-indefinite, not necessarily semi-bounded quadratic forms are revisited. New straightforward proofs of these theorems are given. A number of necessary and sufficient conditions ensuring…
We show that identities involving trigonometric sums recently proved by Harshitha, Vasuki and Yathirajsharma, using Ramanujan's theory of theta functions, were either already in the literature or can be proved easily by adapting results…
In this paper, we reconstruct Euclid's theory of similar triangles, as developed in Book VI of the \textit{Elements}, along with its 20th-century counterparts, formulated within the systems of Hilbert, Birkhoff, Borsuk and Szmielew, Millman…
Previously the authors proved subconvexity of Shintani's zeta function enumerating class numbers of binary cubic forms. Here we return to prove subconvexity of the Maass form twisted version.
In this article we'll present an elementary proof of a theorem of Alexandru Pantazi (1896-1948), Romanian mathematician, regarding the bi-orthological triangles.
The aim of this note is to give two new conceptual proofs of Ionescu-Weitzenb\"ock's inequality. The first one, which is a vector proof, provides us a geometric interpretation of the difference between the two sides of this inequality and…
We provide a new formulation and proof of the triangle altitudes theorem in hyperbolic plane geometry, together with an easily computed discriminant to distinguish between different basic configurations of the altitudes of such a triangle.