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The Calculus of Variations
Published in Steven G. Krantz, Differential Equations, 2022
Show that the triangle with greatest area A for a given perimeter is equilateral. Hint: If x, y, and z are the sides, then A=s(s−x)(s−y)(s−z), where s=(x+y+z)/2.
Angles and triangles
Published in John Bird, Bird's Basic Engineering Mathematics, 2021
An equilateral triangle is one in which all the sides and all the angles are equal; i.e. each is 60∘. An example is shown in triangle ABC in Fig. 21.16(a).
Geometry
Published in Dan Zwillinger, CRC Standard Mathematical Tables and Formulas, 2018
A triangle is equilateral if all of its sides have the same length, or, equivalently, if all of its angles are the same (and equal to 60o). It is isosceles if two sides are the same, or, equivalently, if two angles are the same. Otherwise it is scalene.
Describing the geometric difference of architectural forms in three primary shapes of circle, triangle and square
Published in Journal of Asian Architecture and Building Engineering, 2022
An equilateral triangle is a three-sided plane with all sides equal and equiangular. A building facade in a triangular shape with the vertex pointing upwards converges the most at the apex, facilitating stability and reliability in building design. Heavy base and the pinnacle on the top in many A-framed residential homes enable handling weight because of how the energy is distributed throughout the triangle; the symmetry in the equilateral triangle and the isosceles triangle aids in distributing weight. Trusses exist as bearing structures because of the inherent triangles provided by struts and other bracing elements (Lenartowicz 2014). The Pyramid in Giza, composed of four equilateral triangular faces, has stood for over 4,000 years. The equilateral triangle encloses the smallest area in the most extended perimeter and is a basic unit representing the three-dimensional surface. On the other hand, it can be seen that the triangular-shaped floor plan is unpractical from a functional point of view, as walls converging at an acute angle of 60° contain narrow leftover margins.
Illogical use of the converse of a theorem that can cause an incorrect solution
Published in International Journal of Mathematical Education in Science and Technology, 2022
Triangle BAC given in the exam problem is an isosceles triangle (AB = AC). Solving paragraph c.1 of the exam problem leads to ∠A = 60°, that is triangle BAC is equilateral (see Figure 8). Since in this case point A is on the boundary of a disk with center D and radius t, the midline DE is the only segment of length t.
The problem and geometric application of infinite sequences formed from three given numbers by calculating their pairwise means
Published in International Journal of Mathematical Education in Science and Technology, 2022
Thus triangles with side lengths will tend to an equilateral triangle and their areas form a monotonic increasing sequence. This is consistent with the well-known fact that among all triangles with a given perimeter, an equilateral triangle has the maximum area.