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The golden ratio also appears in hyperbolic geometry, as the maximum distance from a point on one side of an ideal triangle to the closer of the other two sides: this distance, the side length of the equilateral triangle formed by the points of tangency of a circle inscribed within the ideal triangle, is .
In coordinate geometry, the Section formula is a formula used to find the ratio in which a line segment is divided by a point internally or externally. [1] It is used to find out the centroid, incenter and excenters of a triangle. In physics, it is used to find the center of mass of systems, equilibrium points, etc. [2] [3] [4] [5]
Perhaps the most popular among the ortogons is the auron or golden rectangle, which is produced by projecting the diagonal that goes from the middle point of a side of a square to one of the opposite vertexes, until it is aligned with the middle point. Four of these orthogons are harmonic rectangles: the diagon or root-2 rectangle is produced ...
The cross-ratio is a projective invariant in the sense that it is preserved by the projective transformations of a projective line. In particular, if four points lie on a straight line in then their cross-ratio is a well-defined quantity, because any choice of the origin and even of the scale on the line will yield the same value of the cross ...
Since 5 is a prime number there is one subgroup with dihedral symmetry: Dih 1, and 2 cyclic group symmetries: Z 5, and Z 1. These 4 symmetries can be seen in 4 distinct symmetries on the pentagon. John Conway labels these by a letter and group order. [10] Full symmetry of the regular form is r10 and no symmetry is labeled a1.
The four sides can be split into two pairs of adjacent equal-length sides. [7] One diagonal crosses the midpoint of the other diagonal at a right angle, forming its perpendicular bisector. [9] (In the concave case, the line through one of the diagonals bisects the other.) One diagonal is a line of symmetry.
Knowledge of a Line symmetry can be used to simplify an ordinary differential equation through reduction of order. [8] For ordinary differential equations, knowledge of an appropriate set of Lie symmetries allows one to explicitly calculate a set of first integrals, yielding a complete solution without integration.
A circle is thus said to be symmetric under rotation or to have rotational symmetry. If the isometry is the reflection of a plane figure about a line, then the figure is said to have reflectional symmetry or line symmetry; [3] it is also possible for a figure/object to have more than one line of symmetry. [4]