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An equilateral triangle is a triangle that has three equal sides. It is a special case of an isosceles triangle in the modern definition, stating that an isosceles triangle is defined at least as having two equal sides. [1] Based on the modern definition, this leads to an equilateral triangle in which one of the three sides may be considered ...
The Tetractys [also known as the decad] is an equilateral triangle formed from the sequence of the first ten numbers aligned in four rows. It is both a mathematical idea and a metaphysical symbol that embraces within itself—in seedlike form—the principles of the natural world, the harmony of the cosmos, the ascent to the divine, and the ...
Triangles have many types based on the length of the sides and the angles. A triangle whose sides are all the same length is an equilateral triangle, [3] a triangle with two sides having the same length is an isosceles triangle, [4] [a] and a triangle with three different-length sides is a scalene triangle. [7]
For any interior point P, the sum of the lengths of the perpendiculars s + t + u equals the height of the equilateral triangle.. Viviani's theorem, named after Vincenzo Viviani, states that the sum of the shortest distances from any interior point to the sides of an equilateral triangle equals the length of the triangle's altitude. [1]
The pedal triangle of an isodynamic point, the triangle formed by dropping perpendiculars from to each of the three sides of triangle , is equilateral, [5] as is the triangle formed by reflecting across each side of the triangle. [7] Among all the equilateral triangles inscribed in triangle , the pedal triangle of the first isodynamic point is ...
The dihedral group D 3 is the symmetry group of an equilateral triangle, that is, it is the set of all rigid transformations (reflections, rotations, and combinations of these) that leave the shape and position of this triangle fixed.
A deltahedron is a polyhedron whose faces are all equilateral triangles. The deltahedron is named by Martyn Cundy , after the Greek capital letter delta resembling a triangular shape Δ. [ 1 ] The deltahedron can be categorized by the property of convexity .
Twelve key lengths of a triangle are the three side lengths, the three altitudes, the three medians, and the three angle bisectors. Together with the three angles, these give 95 distinct combinations, 63 of which give rise to a constructible triangle, 30 of which do not, and two of which are underdefined. [13]: pp. 201–203