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Angles, Area, and Perimeter Caught in a Cubic - mklabgr - 09-07-2026
Angles, Area, and Perimeter Caught in a Cubic
Authors: George Baloglou & Michel Helfgott
Published:Forum Geometricorum, Vol. 8 (2008), pp. 13–25
The paper studies how much the angles and side lengths of a triangle are constrained when its area $A$ and perimeter $P$ are fixed. The key observation is that the problem naturally leads to cubic equations. For an isosceles triangle with base $x$, area $A$, and perimeter $P$, the base must satisfy $2Px^3-P^2x^2+16A^2=0$. Analyzing this cubic shows that there are exactly two distinct isosceles triangles with the given $A$ and $P$ whenever $P^2>12\sqrt{3},A$, while equality $P^2=12\sqrt{3},A$ produces the unique equilateral triangle. This also gives a geometric proof of the classical triangular isoperimetric inequality $P^2\geq12\sqrt{3},A$, with equality only for the equilateral triangle.
The authors then develop a generalization of Newton's parametrization, expressing the three side lengths in terms of $A$, $P$, and one angle $\phi$. Their main angular result states that every angle of a non-equilateral triangle with fixed area and perimeter lies between the vertex angles $\phi_1$ and $\phi_2$ of the two associated isosceles triangles: $\phi_1\leq\phi\leq\phi_2$, where $\phi_1<\frac{\pi}{3}<\phi_2$. Thus the extremal possible angles always occur in isosceles triangles. For the illustrative case $A=3$ and $P=10$, all angles must lie between approximately $19.003^\circ$ and $122.351^\circ$.
Finally, the authors extend the cubic method to bound ratios of sides. By introducing a prescribed ratio $r=z/y$, they obtain another cubic and reinterpret Heron's formula geometrically through what they call Heron's curve. Tangent lines to this curve determine the sharpest possible side ratios. For $A=3$ and $P=10$, every ratio between two sides satisfies approximately $0.3273\leq r\leq3.0551$. The extreme triangle has sides approximately ${4.2048,4.3661,1.4291}$. The paper therefore connects elementary triangle geometry, cubic equations, Newton's formulas, Heron's formula, and optimization into a unified framework.
ARTICLE [PDF]