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In the special cases of one of the diagonals or sides being a diameter of the circle, this theorem gives rise directly to the angle sum and difference trigonometric identities. [17] The relationship follows most easily when the circle is constructed to have a diameter of length one, as shown here.
[2] An inversion centered at p transforms A and B into concentric circles. [3] The midpoint of the two limiting points is the point where the radical axis of A and B crosses the line through their centers. This intersection point has equal power distance to all the circles in the pencil containing A and B. The limiting points themselves can be ...
The distances between the centers of the nearer and farther circles, O 2 and O 1 and the point where the two outer tangents of the two circles intersect (homothetic center), S respectively can be found out using similarity as follows: Here, r can be r 1 or r 2 depending upon the need to find distances from the centers of the nearer or farther ...
Monge's theorem states that the three such points given by the three pairs of circles always lie in a straight line. In the case of two of the circles being of equal size, the two external tangent lines are parallel. In this case Monge's theorem asserts that the other two intersection points must lie on a line parallel to those two external ...
Such polygons may have any number of sides greater than 1. Two-sided spherical polygons—lunes, also called digons or bi-angles—are bounded by two great-circle arcs: a familiar example is the curved outward-facing surface of a segment of an orange. Three arcs serve to define a spherical triangle, the principal subject of this article.
In trigonometry, the law of tangents or tangent rule [1] is a statement about the relationship between the tangents of two angles of a triangle and the lengths of the opposing sides. In Figure 1, a, b, and c are the lengths of the three sides of the triangle, and α, β, and γ are the angles opposite those three respective sides.
The eyeball theorem was discovered in 1960 by the Peruvian mathematician Antonio Gutierrez. [2] However without the use of its current name it was already posed and solved as a problem in an article by G. W. Evans in 1938. [3] Furthermore Evans stated that problem was given in an earlier examination paper. [4]
The number of points (n), chords (c) and regions (r G) for first 6 terms of Moser's circle problem. In geometry, the problem of dividing a circle into areas by means of an inscribed polygon with n sides in such a way as to maximise the number of areas created by the edges and diagonals, sometimes called Moser's circle problem (named after Leo Moser), has a solution by an inductive method.