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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]
The intercept theorem, also known as Thales's theorem, basic proportionality theorem or side splitter theorem, is an important theorem in elementary geometry about the ratios of various line segments that are created if two rays with a common starting point are intercepted by a pair of parallels.
The fixed points of the 3-cycles are exp(±iπ/3), corresponding under M to the poles of the sphere: exp(iπ/3) is the origin and exp(−iπ/3) is the point at infinity. Each 3 -cycle is a 1/3 turn rotation about their axis, and they are exchanged by the 2 -cycles.
The eccentricity is also the ratio of the semimajor axis a to the distance d from the center to the directrix: e = a d . {\displaystyle e={\frac {a}{d}}.} The eccentricity can be expressed in terms of the flattening f (defined as f = 1 − b / a {\displaystyle f=1-b/a} for semimajor axis a and semiminor axis b ):
The semi-minor axis of an ellipse runs from the center of the ellipse (a point halfway between and on the line running between the foci) to the edge of the ellipse. The semi-minor axis is half of the minor axis. The minor axis is the longest line segment perpendicular to the major axis that connects two points on the ellipse's edge.
Equivalently, the tangents of the ellipsoid containing point V are the lines of a circular cone, whose axis of rotation is the tangent line of the hyperbola at V. [ 15 ] [ 16 ] If one allows the center V to disappear into infinity, one gets an orthogonal parallel projection with the corresponding asymptote of the focal hyperbola as its direction.
Determination of a circle, that intersects four circles by the same angle. [2] Solving the Problem of Apollonius; Construction of the Malfatti circles: [3] For a given triangle determine three circles, which touch each other and two sides of the triangle each. Spherical version of Malfatti's problem: [4] The triangle is a spherical one.
If p is a point not on a straight with harmonic points, the joins of p with the points are harmonic straights. Similarly, if the axis of a pencil of planes is skew to a straight with harmonic points, the planes on the points are harmonic planes. [6] A set of four in such a relation has been called a harmonic quadruple. [7]