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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.
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 ...
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. Extend DB past B and find the intersection of DB and the circle BC, labeled E. Create a circle centered at D and passing through E (the blue circle).
A compass-only construction of doubling the length of segment AB. Given a line segment AB find a point C on the line AB such that B is the midpoint of line segment AC. [10] Construct point D as the intersection of circles A(B) and B(A). (∆ABD is an equilateral triangle.) Construct point E ≠ A as the intersection of circles D(B) and B(D).
Thus one only has to find a compass and straightedge construction for n-gons where n is a Fermat prime. 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 ...
In geometry, Lemoine's problem is a straightedge and compass construction problem posed by French mathematician Émile Lemoine in 1868: [1] [2]. Given one vertex of each of the equilateral triangles placed on the sides of a triangle, construct the original triangle.
Hutcheson constructed a cylinder from the angle to be trisected by drawing an arc across the angle, completing it as a circle, and constructing from that circle a cylinder on which a, say, equilateral triangle was inscribed (a 360-degree angle divided in three).
The square root of 2 is equal to the length of the hypotenuse of a right triangle with legs of length 1 and is therefore a constructible number. In geometry and algebra, a real number is constructible if and only if, given a line segment of unit length, a line segment of length | | can be constructed with compass and straightedge in a finite number of steps.