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Special cases of Apollonius' problem are those in which at least one of the given circles is a point or line, i.e., is a circle of zero or infinite radius. The nine types of such limiting cases of Apollonius' problem are to construct the circles tangent to: three points (denoted PPP, generally 1 solution)
A unit vector means that the vector has a length of 1, which is also known as normalized. Orthogonal means that the vectors are all perpendicular to each other. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of unit length. An orthonormal set which forms a basis is called an orthonormal basis.
The straight lines which form right angles are called perpendicular. [8] Euclid uses right angles in definitions 11 and 12 to define acute angles (those smaller than a right angle) and obtuse angles (those greater than a right angle). [9] Two angles are called complementary if their sum is a right angle. [10]
Then the vertex of this angle is X and traces out the pedal curve. As the angle moves, its direction of motion at P is parallel to PX and its direction of motion at R is parallel to the tangent T = RX. Therefore, the instant center of rotation is the intersection of the line perpendicular to PX at P and perpendicular to RX at R, and this point ...
adventitious quadrangles problem. A quadrilateral such as BCEF is called an adventitious quadrangle when the angles between its diagonals and sides are all rational angles, angles that give rational numbers when measured in degrees or other units for which the whole circle is a rational number. Numerous adventitious quadrangles beyond the one ...
The line segments AB and CD are perpendicular to each other. In mathematics, orthogonality is the generalization of the geometric notion of perpendicularity.Although many authors use the two terms perpendicular and orthogonal interchangeably, the term perpendicular is more specifically used for lines and planes that intersect to form a right angle, whereas orthogonal is used in generalizations ...
The reason is that the value of sine for the angle of the triangle does not uniquely determine this angle. For example, if sin β = 0.5, the angle β can equal either 30° or 150°. Using the law of cosines avoids this problem: within the interval from 0° to 180° the cosine value unambiguously determines its angle.
If ^ is an acute angle and AB is any segment, then there exists a point P on the ray and a point Q on the ray , such that PQ is perpendicular to OX and PQ > AB. Aristotle's axiom is a consequence of the Archimedean property , [ 1 ] and the conjunction of Aristotle's axiom and the Lotschnittaxiom , which states that "Perpendiculars raised on ...