Editing Perpendicular (section)

===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.