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Construction of equilateral triangle with compass and straightedge. An equilateral triangle is easily constructed using a straightedge and compass, because 3 is a Fermat prime. Draw a straight line, and place the point of the compass on one end of the line, and swing an arc from that point to the other point of the line segment.
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] It is a theorem commonly employed in various math competitions, secondary school mathematics examinations, and has wide applicability to ...
Ptolemy's Theorem yields as a corollary a pretty theorem [2] regarding an equilateral triangle inscribed in a circle. Given An equilateral triangle inscribed on a circle and a point on the circle. The distance from the point to the most distant vertex of the triangle is the sum of the distances from the point to the two nearer vertices.
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.
Heron's formula. A triangle with sides a, b, and c. In geometry, Heron's formula (or Hero's formula) gives the area of a triangle in terms of the three side lengths Letting be the semiperimeter of the triangle, the area is [1] It is named after first-century engineer Heron of Alexandria (or Hero) who ...
Fig 1. Construction of the first isogonic center, X(13). When no angle of the triangle exceeds 120°, this point is the Fermat point. In Euclidean geometry, the Fermat point of a triangle, also called the Torricelli point or Fermat–Torricelli point, is a point such that the sum of the three distances from each of the three vertices of the triangle to the point is the smallest possible [1] or ...
Main parameters and notation. The parameters most commonly appearing in triangle inequalities are: the side lengths a, b, and c; the semiperimeter s = (a + b + c) / 2 (half the perimeter p); the angle measures A, B, and C of the angles of the vertices opposite the respective sides a, b, and c (with the vertices denoted with the same symbols as ...
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.
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