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Hyperbola: the midpoints of parallel chords lie on a line. Hyperbola: the midpoint of a chord is the midpoint of the corresponding chord of the asymptotes. The midpoints of parallel chords of a hyperbola lie on a line through the center (see diagram). The points of any chord may lie on different branches of the hyperbola.
A parallel ruler can be used to draw a line through a given point A and parallel to a given ray a [3]. For any two lines, a hyperbolic ruler can be used to construct a line that is parallel to the first line and perpendicular to the second. [3] A few notes on the uses of rulers are:
Lines through a given point P and asymptotic to line R. Non-intersecting lines in hyperbolic geometry also have properties that differ from non-intersecting lines in Euclidean geometry: For any line R and any point P which does not lie on R, in the plane containing line R and point P there are at least two distinct lines through P that do not ...
Considering the pencils of confocal ellipses and hyperbolas (see lead diagram) one gets from the geometrical properties of the normal and tangent at a point (the normal of an ellipse and the tangent of a hyperbola bisect the angle between the lines to the foci). Any ellipse of the pencil intersects any hyperbola orthogonally (see diagram).
Consider the points at which the straight line drawn through E parallel to AB intersects the conic a second time to be the sum of the points A and B. For the hyperbola = with the fixed point E = (1,0) the sum of the points (, ) and (, ) is the point (+, +) under the parametrization = and = this addition corresponds to the addition of ...
Lines perpendicular to line l are modeled by chords whose extension passes through the pole of l. Hence we draw the unique line between the poles of the two given lines, and intersect it with the boundary circle; the chord of intersection will be the desired common perpendicular of the ultraparallel lines.
Specifically all the points lying on the line have their isogonal conjugates lying on the hyperbola. The Nagel point lies on the curve since its isogonal conjugate is the point of concurrency of the lines joining the vertices and the opposite Mixtilinear incircle touchpoints, also the in-similitude of the incircle and the circumcircle.
A hyperbola and its conjugate may be constructed as conic sections obtained from an intersecting plane that meets tangent double cones sharing the same apex. Each cone has an axis, and the plane section is parallel to the plane formed by the axes. Using analytic geometry, the hyperbolas satisfy the symmetric equations