Editing Perpendicular (section)

==In circles and other conics==

===Circles===

Each [[diameter]] of a [[circle]] is perpendicular to the [[tangent line]] to that circle at the point where the diameter intersects the circle.

A line segment through a circle's center bisecting a [[chord (geometry)|chord]] is perpendicular to the chord.

If the intersection of any two perpendicular chords divides one chord into lengths ''a'' and ''b'' and divides the other chord into lengths ''c'' and ''d'', then {{nowrap|''a''<sup>2</sup> + ''b''<sup>2</sup> + ''c''<sup>2</sup> + ''d''<sup>2</sup>}} equals the square of the diameter.<ref>Posamentier and Salkind, ''Challenging Problems in Geometry'', Dover, 2nd edition, 1996: pp. 104–105, #4–23.</ref>

The sum of the squared lengths of any two perpendicular chords intersecting at a given point is the same as that of any other two perpendicular chords intersecting at the same point, and is given by 8''r''<sup>2</sup> – 4''p''<sup>2</sup> (where ''r'' is the circle's radius and ''p'' is the distance from the center point to the point of intersection).<ref>''[[College Mathematics Journal]]'' 29(4), September 1998, p. 331, problem 635.</ref>

[[Thales' theorem]] states that two lines both through the same point on a circle but going through opposite endpoints of a diameter are perpendicular. This is equivalent to saying that any diameter of a circle subtends a right angle at any point on the circle, except the two endpoints of the diameter.

===Ellipses===

The major and minor [[axis of symmetry|axes]] of an [[ellipse]] are perpendicular to each other and to the tangent lines to the ellipse at the points where the axes intersect the ellipse.

The major axis of an ellipse is perpendicular to the [[directrix (conic section)|directrix]] and to each [[latus rectum]].

===Parabolas===

In a [[parabola]], the axis of symmetry is perpendicular to each of the latus rectum, the directrix, and the tangent line at the point where the axis intersects the parabola.

From a point on the tangent line to a parabola's vertex, the [[Parabola#Intersection of a tangent and perpendicular from focus|other tangent line to the parabola]] is perpendicular to the line from that point through the parabola's [[focus (geometry)|focus]].

The [[Parabola#Orthoptic property|orthoptic property]] of a parabola is that If two tangents to the parabola are perpendicular to each other, then they intersect on the directrix. Conversely, two tangents which intersect on the directrix are perpendicular. This implies that, seen from any point on its directrix, any parabola subtends a right angle.

===Hyperbolas===

The [[hyperbola#Equation|transverse axis]] of a [[hyperbola]] is perpendicular to the conjugate axis and to each directrix.

The product of the perpendicular distances from a point P on a hyperbola or on its conjugate hyperbola to the asymptotes is a constant independent of the location of P.

A [[hyperbola#Rectangular hyperbola|rectangular hyperbola]] has [[asymptote]]s that are perpendicular to each other. It has an [[eccentricity (mathematics)|eccentricity]] equal to <math>\sqrt{2}.</math>