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  2. Hyperbola - Wikipedia

    en.wikipedia.org/wiki/Hyperbola

    For a hyperbola =, > the intersection points of orthogonal tangents lie on the circle + =. This circle is called the orthoptic of the given hyperbola. The tangents may belong to points on different branches of the hyperbola.

  3. Tangent lines to circles - Wikipedia

    en.wikipedia.org/wiki/Tangent_lines_to_circles

    The intersection points T 1 and T 2 of the circle C and the new circle are the tangent points for lines passing through P, by the following argument. The line segments OT 1 and OT 2 are radii of the circle C; since both are inscribed in a semicircle, they are perpendicular to the line segments PT 1 and PT 2, respectively. But only a tangent ...

  4. Concurrent lines - Wikipedia

    en.wikipedia.org/wiki/Concurrent_lines

    In a hyperbola the following are concurrent: (1) a circle passing through the hyperbola's foci and centered at the hyperbola's center; (2) either of the lines that are tangent to the hyperbola at the vertices; and (3) either of the asymptotes of the hyperbola.

  5. Orthoptic (geometry) - Wikipedia

    en.wikipedia.org/wiki/Orthoptic_(geometry)

    An isoptic is the set of points for which two tangents of a given curve meet at a fixed angle (see below). An isoptic of two plane curves is the set of points for which two tangents meet at a fixed angle. Thales' theorem on a chord PQ can be considered as the orthoptic of two circles which are degenerated to the two points P and Q.

  6. Confocal conic sections - Wikipedia

    en.wikipedia.org/wiki/Confocal_conic_sections

    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).

  7. Conic section - Wikipedia

    en.wikipedia.org/wiki/Conic_section

    If the intersection point is double, the line is a tangent line. Intersecting with the line at infinity, each conic section has two points at infinity. If these points are real, the curve is a hyperbola; if they are imaginary conjugates, it is an ellipse; if there is only one double point, it is a parabola.

  8. Intersection (geometry) - Wikipedia

    en.wikipedia.org/wiki/Intersection_(geometry)

    Intersection of two line segments. For two non-parallel line segments (,), (,) and (,), (,) there is not necessarily an intersection point (see diagram), because the intersection point (,) of the corresponding lines need not to be contained in the line segments.

  9. Dupin indicatrix - Wikipedia

    en.wikipedia.org/wiki/Dupin_indicatrix

    In particular, the indicatrix of an umbilical point is a circle. For hyperbolic points, where the Gaussian curvature is negative, the intersection will form a hyperbola. Two different hyperbolas will be formed on either side of the tangent plane. These hyperbolas share the same axis and asymptotes.