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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.
Brahmagupta's formula; Bretschneider's formula; Compass and straightedge constructions. Squaring the circle; Complex geometry; Conic section. Focus; Circle. List of circle topics; Thales' theorem; Circumcircle; Concyclic; Incircle and excircles of a triangle; Orthocentric system; Monge's theorem; Power center; Nine-point circle; Circle points ...
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 ...
A stone pillar in the 6th-century Marundeeswarar Temple in India shows three equilateral triangles rotated from each other to form a regular enneagram; like the Borromean rings these three triangles are linked and not pairwise linked, [12] but this crossing pattern describes a different link than the Borromean rings. [13]
For example, AF / FB is defined as having positive value when F is between A and B and negative otherwise. Ceva's theorem is a theorem of affine geometry , in the sense that it may be stated and proved without using the concepts of angles, areas, and lengths (except for the ratio of the lengths of two line segments that are collinear ).
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