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  2. Angle bisector theorem - Wikipedia

    en.wikipedia.org/wiki/Angle_bisector_theorem

    The angle bisector theorem is commonly used when the angle bisectors and side lengths are known. It can be used in a calculation or in a proof. An immediate consequence of the theorem is that the angle bisector of the vertex angle of an isosceles triangle will also bisect the opposite side.

  3. Constructions in hyperbolic geometry - Wikipedia

    en.wikipedia.org/wiki/Constructions_in...

    Bisect one of the angles made by these two lines and name the angle bisector b. Using a hyperbolic ruler, construct a line c such that c is perpendicular to b and parallel to a. As a result, c is also parallel to a', making c the common parallel to lines a and a'. [3] Case 2: a and a' are parallel to each other

  4. Bisection - Wikipedia

    en.wikipedia.org/wiki/Bisection

    To bisect an angle with straightedge and compass, one draws a circle whose center is the vertex. The circle meets the angle at two points: one on each leg. Using each of these points as a center, draw two circles of the same size. The intersection of the circles (two points) determines a line that is the angle bisector.

  5. Hyperbolic geometry - Wikipedia

    en.wikipedia.org/wiki/Hyperbolic_geometry

    The side and angle bisectors will, depending on the side length and the angle between the sides, be limiting or diverging parallel. If the bisectors are limiting parallel then it is an apeirogon and can be inscribed and circumscribed by concentric horocycles .

  6. Concurrent lines - Wikipedia

    en.wikipedia.org/wiki/Concurrent_lines

    A convex quadrilateral is ex-tangential if and only if there are six concurrent angles bisectors: the internal angle bisectors at two opposite vertex angles, the external angle bisectors at the other two vertex angles, and the external angle bisectors at the angles formed where the extensions of opposite sides intersect.

  7. Special cases of Apollonius' problem - Wikipedia

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

    The intersection points of this circle with the two given lines (5) are T1 and T2. Two circles of the same radius, centered on T1 and T2, intersect at points P and Q. The line through P and Q (1) is an angle bisector. Rays have one angle bisector; lines have two, perpendicular to one another.

  8. Thales's theorem - Wikipedia

    en.wikipedia.org/wiki/Thales's_theorem

    adjacent angles in a parallelogram are supplementary (add to 180°) and, the diagonals of a rectangle are equal and cross each other in their median point. Let there be a right angle ∠ ABC, r a line parallel to BC passing by A, and s a line parallel to AB passing by C. Let D be the point of intersection of lines r and s.

  9. Angle of parallelism - Wikipedia

    en.wikipedia.org/wiki/Angle_of_parallelism

    The angle depends on the segment length a between the right angle and the vertex of the angle of parallelism. Given a point not on a line, drop a perpendicular to the line from the point. Let a be the length of this perpendicular segment, and Π ( a ) {\displaystyle \Pi (a)} be the least angle such that the line drawn through the point does not ...