Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2603.00001

Generalized Chapple--Euler Relation

Vladimir Dragović·Mohammad Hassan Murad
View original source ↗Read source document ↗Export metadata ↓

Abstract

We provide a new proof of the necessary and sufficient condition for a triangle to be circumscribed about a central conic (ellipse or hyperbola), expressed in terms of the circumradius and the distances from the circumcenter to the foci. If the inscribed conic is an ellipse, in the limiting case, where the foci coincide, the condition reduces to the classical Chapple--Euler relation. We also prove that the sum of the squares of the sides of a triangle in a family inscribed in a circle and circumscribed about a central conic remains invariant throughout the family if and only if the center of the circle coincides either with the center of the conic or with one of its foci. Using Blaschke products of degree three and affine transformations, we characterize all central $3$-Poncelet pairs with constant triangle area. We prove that the associated family of triangles has constant area if and only if the two conics are homothetic ellipses. We also prove two observations of Reznik concerning the invariance of the total area of the power circles of Poncelet triangles by relating them to Apollonius's identity on medians and establish a more general result valid for both ellipses and hyperbolas. Finally, we propose two conjectures on area invariance of power-circles.

Explore related subjects

math.GM

Keep this discovery

BibTeXRIS

Vladimir Dragović, Mohammad Hassan Murad. 2025-12-25. Generalized Chapple--Euler Relation. https://doi.org/10.1007/s40879-026-00919-z

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Average Chord Lengths in a Triangle

Let $P$ be a point inside a triangle $T$. We consider the average length of the chords of $T$ through $P$, where the direction of the chord is chosen uniformly. An elementary formula is obtained in terms of the distances from $P$ to the sides and vertices of the triangle. Several classical triangle centers give especially simple specializations. For example, if $I$ is the incenter, then \[ M_T(I)=\frac{2r}{\pi} \log\left(\cot\frac A4\cot\frac B4\cot\frac C4\right). \] Our main result is the sharp inequality \[ M_T(P)\le \frac{p}{\pi\sqrt3}\log(2+\sqrt3), \] valid simultaneously for every triangle of perimeter $p$ and every interior point $P$. Thus, among all such pairs $(T,P)$, the largest possible average chord length occurs only when $T$ is equilateral and $P$ is its center. The proof is an elementary symmetrization argument. We close with brief remarks relating the problem to the radial center of a convex body, the electrostatic potential center of a triangle, and dual quermassintegrals.

math.GM↗
arXiv2026

A Proof of Liu's Conjecture on the Fundamental Triangle Inequality

Let $a,b,c$ be the side lengths of a triangle, and let $R$ and $r$ denote its circumradius and inradius, respectively. We prove a conjecture of Liu stating that \[\sum_{\mathrm{cyc}} \left(\frac{a(b+c-a)}{bc}\right)^k \geq 2+\left(\frac{2r}{R}\right)^k,~~k>1, \] with the reverse inequality for $0<k<1$. The proof reduces the problem to three positive variables with fixed sum and product. We also determine the equality cases.

math.GM↗