Geometry

By using a Morse function and a Witten deformation argument, we obtain an upper bound for the dimensions of the space of divergence-free symmetric Killing p-tensors on a closed Riemannian manifold and explicitly calculate it for $p=2$.

Let $▵ABC$ be a triangle in the plane $\mathcal{E}$, K be its symmedian point, and $\mathcal{C}$ be its circumcircle. Assume that P is a point on $\mathcal{C}$ such that it is not A, B, or C, it does not lie on the medians of $▵ABC$, and the lines ${\ell }_{AP}$, ${\ell }_{BP}$, and ${\ell }_{CP}$ intersect ${\ell }_{BC}$, ${\ell }_{CA}$, and ${\ell }_{AB}$ at points $P_a$, $P_b$, and $P_c$, respectively. By the Ceva Concurrence Theorem, the harmonic conjugates $Q_a$, $Q_b$, and $Q_c$ of $P_a$, $P_b$, and $P_c$ in $\overline{BC}$, $\overline{CA}$, and $\overline{AB}$, respectively, are collinear. We prove that K lies on the line through $Q_a$, $Q_b$, and $Q_c$, which provides a new characterization of the symmedian point. Moreover, this one-to-one correspondence extends to a bijection from the entire circumcircle $\mathcal{C}$ onto the set ${\mathcal{L}}_K$ of all the lines in $\mathcal{E}$ that pass through K.

It is well known among geometry scholars that the golden triangle, an isosceles triangle with sides and base in golden ratio, maintains a significant relationship with regular polygons, notably the regular pentagon, pentagram, and decagon. Extensive mathematical literature addresses this subject. Furthermore, its close association with the golden ratio—a mathematical concept describing a harmonious and proportionate relationship between segments—renders it a noteworthy element in the fields of geometry, art, and architecture. Nevertheless, the interrelationships among these mathematical constructs frequently reveal unexpected configurations, thereby accentuating intriguing patterns. The purpose of this investigation is to highlight these novel configurations, which indicate new connections between the golden triangle and regular polygons.