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  2. Special cases of Apollonius' problem - Wikipedia

    en.wikipedia.org/wiki/Special_cases_of_Apollonius...

    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)

  3. Langley's Adventitious Angles - Wikipedia

    en.wikipedia.org/wiki/Langley's_Adventitious_Angles

    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 ...

  4. Orthonormality - Wikipedia

    en.wikipedia.org/wiki/Orthonormality

    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.

  5. Angle trisection - Wikipedia

    en.wikipedia.org/wiki/Angle_trisection

    Three problems proved elusive, specifically, trisecting the angle, doubling the cube, and squaring the circle. The problem of angle trisection reads: Construct an angle equal to one-third of a given arbitrary angle (or divide it into three equal angles), using only two tools: an unmarked straightedge, and; a compass.

  6. Aristotle's axiom - Wikipedia

    en.wikipedia.org/wiki/Aristotle's_axiom

    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 ...

  7. Pedal curve - Wikipedia

    en.wikipedia.org/wiki/Pedal_curve

    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 ...

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