orthogonal circles


Two circles intersecting orthogonally (http://planetmath.org/ConvexAngle) are orthogonal curves and called orthogonal circlesMathworldPlanetmath of each other.

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Since the tangent of circle is perpendicularMathworldPlanetmathPlanetmathPlanetmath to the radius drawn to the tangency point, the both radii of two orthogonal circles drawn to the point of intersectionMathworldPlanetmathPlanetmath and the line segmentMathworldPlanetmath connecting the centres form a right triangleMathworldPlanetmath. If  (x-a1)2+(y-b1)2=r12  and  (x-a2)2+(y-b2)2=r22  are the equations of the circles, then, by Pythagorean theoremMathworldPlanetmathPlanetmath,

r12+r22=(a2-a1)2+(b2-b1)2 (1)

is the condition of the orthogonality (http://planetmath.org/OrthogonalCurves) of those circles.

The equation (1) tells that the centre of one circle is always outside its orthogonal circle.  If  (x0,y0)  is an arbitrary point outside the circle  (x-a)2+(y-b)2=r2,  one can always draw with that point as centre the orthogonal circle of this circle:  its radius is the limited tangent from  (x0,y0)  to the given circle. The square (http://planetmath.org/SquareOfANumber) of the limited tangent is equal to the power of the point with respect to the circle and thus  (x0-a)2+(y0-b)2-r2.  Accordingly, the equation of the orthogonal circle is

(x-x0)2+(y-y0)2=(x0-a)2+(y0-b)2-r2.

One of two orthogonalMathworldPlanetmathPlanetmathPlanetmath (http://planetmath.org/Orthogonal) circles harmonically any diameterMathworldPlanetmath of the other circle.

Title orthogonal circles
Canonical name OrthogonalCircles
Date of creation 2013-03-22 17:41:32
Last modified on 2013-03-22 17:41:32
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 7
Author pahio (2872)
Entry type Topic
Classification msc 51N20
Related topic PerpendicularityInEuclideanPlane
Defines orthogonal circle