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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.
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.
Analogue on a Cartesian grid by adding lines of slope −1. The scale of the c axis is that of the a and b axes. The cross denotes the point a = b = c. Cartesian coordinates are useful for plotting points in the triangle. Consider an equilateral ternary plot where a = 100% is placed at (x,y) = (0,0) and b = 100% at (1,0).
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.
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 ...
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 ).
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.
draw an equilateral triangle that has the middle segment from step 1 as its base and points outward. remove the line segment that is the base of the triangle from step 2. The first iteration of this process produces the outline of a hexagram. The Koch snowflake is the limit approached as the above steps are followed indefinitely.
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