7 papers
Siebeck--Marden Theorems and Symmetric Parametrizations of Cyclic Polygons
Mohammad Hassan Murad
The classical Blaschke-product approach provides an elegant description of triangles inscribed in a circle and circumscribed about an ellipse. Motivated by the Siebeck--Marden theo…
A Vector Generalization of Euler's Quadrilateral Theorem
Mohammad Hassan Murad
In this paper, we develop a unified algebraic framework for Euler-type identities in real and complex inner product spaces. Starting from the parallelogram identity, we derive Apol…
On Some Geometric Conjectures for Poncelet Polygons
Mohammad Hassan Murad
In this paper, we generalize some recent observations made by Dan Reznik in a series of computer-assisted experiments. Motivated by the recent proof of one of the original observat…
Generalized Chapple--Euler Relation
Vladimir DragoviÄ, Mohammad Hassan Murad
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 circumra…
A Symmetric Polynomial Approach to Poncelet Triangles Inscribed in a circle and circumscribed about Central Conics
Mohammad Hassan Murad
We develop a symmetric-polynomial framework for the study of one-parameter families of Poncelet triangles. For triangles inscribed in the unit circle and circumscribed about a cent…
Parable of the Parabola
Vladimir DragoviÄ, Mohammad Hassan Murad
We study triangles and quadrilaterals which are inscribed in a circle and circumscribed about a parabola. Although these are particular cases of the celebrated Poncelet's Theorem,…