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Construction of equilateral triangle with compass and straightedge. The equilateral triangle can be constructed in different ways by using circles. The first proposition in the Elements first book by Euclid. Start by drawing a circle with a certain radius, placing the point of the compass on the circle, and drawing another circle with the same ...
If we draw both circles, two new points are created at their intersections. Drawing lines between the two original points and one of these new points completes the construction of an equilateral triangle. Therefore, in any geometric problem we have an initial set of symbols (points and lines), an algorithm, and some results.
A proof from Euclid's Elements that, given a line segment, one may construct an equilateral triangle that includes the segment as one of its sides: an equilateral triangle ΑΒΓ is made by drawing circles Δ and Ε centered on the points Α and Β, and taking one intersection of the circles as the third vertex of the triangle.
The Sierpiński triangle may be constructed from an equilateral triangle by repeated removal of triangular subsets: Start with an equilateral triangle. Subdivide it into four smaller congruent equilateral triangles and remove the central triangle. Repeat step 2 with each of the remaining smaller triangles infinitely.
Given points A, B, and C, construct a circle centered at A with radius the length of BC (that is, equivalent to the solid green circle, but centered at A). Draw a circle centered at A and passing through B and vice versa (the red circles). They will intersect at point D and form the equilateral triangle ABD.
The construction for an equilateral triangle is simple and has been known since antiquity; see Equilateral triangle. Constructions for the regular pentagon were described both by Euclid (Elements, ca. 300 BC), and by Ptolemy (Almagest, ca. 150 AD). Although Gauss proved that the regular 17-gon is constructible, he did not actually show how to ...
The triangle XYZ is called the inner Napoleon triangle of ABC. Napoleon's theorem asserts that this triangle is an equilateral triangle. In Clark Kimberling's Encyclopedia of Triangle Centers, the second Napoleon point is denoted by X(18). [3]
Given a square, construct equilateral triangles on two adjacent edges, either both inside or both outside the square. Then the triangle formed by joining the vertex of the square distant from both triangles and the vertices of the triangles distant from the square is equilateral. [2]