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In architecture, an example can be seen in the cross-section of the Gateway Arch and the surface of the Vegreville egg. [25] [26] It appears in the flag of Nicaragua and the flag of the Philippines. [27] [28] It is a shape of a variety of road signs, including the yield sign. [29] The equilateral triangle occurs in the study of stereochemistry.
A standard example is the Reuleaux triangle, the intersection of three circles, each centered where the other two circles cross. [2] Its boundary curve consists of three arcs of these circles, meeting at 120° angles, so it is not smooth , and in fact these angles are the sharpest possible for any curve of constant width.
Napoleon's theorem: If the triangles centered on L, M, N are equilateral, then so is the green triangle.. In geometry, Napoleon's theorem states that if equilateral triangles are constructed on the sides of any triangle, either all outward or all inward, the lines connecting the centres of those equilateral triangles themselves form an equilateral triangle.
The n-th centered triangular number, corresponding to n layers plus the center, is given by the formula:, = + (+) = + +. Each centered triangular number has a remainder of 1 when divided by 3, and the quotient (if positive) is the previous regular triangular number.
Additionally, among the curves of constant width, the Reuleaux triangle is the one with both the largest and the smallest inscribed equilateral triangles. [15] The largest equilateral triangle inscribed in a Reuleaux triangle is the one connecting its three corners, and the smallest one is the one connecting the three midpoints of its sides ...
Van Schooten's theorem, named after the Dutch mathematician Frans van Schooten, describes a property of equilateral triangles.It states: For an equilateral triangle with a point on its circumcircle the length of longest of the three line segments ,, connecting with the vertices of the triangle equals the sum of the lengths of the other two.
Broken down, 3 6; 3 6 (both of different transitivity class), or (3 6) 2, tells us that there are 2 vertices (denoted by the superscript 2), each with 6 equilateral 3-sided polygons (triangles). With a final vertex 3 4.6, 4 more contiguous equilateral triangles and a single regular hexagon.
If a, b and c each cannot be negative, P is restricted to the triangle bounded by A, B and C, as in (2). In (3), the axes are rotated to give an isometric view. The triangle, viewed face-on, appears equilateral. In (4), the distances of P from lines BC, AC and AB are denoted by a′, b′ and c′, respectively.