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Laguerre defined the power of a point P with respect to an algebraic curve of degree n to be the sum of the distances from the point to the intersections of a circle through the point with the curve, divided by the nth power of the diameter d. Laguerre showed that this number is independent of the diameter (Laguerre 1905).
The value of the two products in the chord theorem depends only on the distance of the intersection point S from the circle's center and is called the absolute value of the power of S; more precisely, it can be stated that: | | | | = | | | | = where r is the radius of the circle, and d is the distance between the center of the circle and the ...
As shown above, if a circle passes through two given points P 1 and P 2, its center must lie somewhere on the perpendicular bisector line of the two points. Therefore, if the solution circle passes through three given points P 1, P 2 and P 3, its center must lie on the perpendicular bisectors of ¯, ¯ and ¯. At least two of these bisectors ...
Equal chords are subtended by equal angles from the center of the circle. A chord that passes through the center of a circle is called a diameter and is the longest chord of that specific circle. If the line extensions (secant lines) of chords AB and CD intersect at a point P, then their lengths satisfy AP·PB = CP·PD (power of a point theorem).
Gauss's circle problem asks how many points there are inside this circle of the form (,) where and are both integers. Since the equation of this circle is given in Cartesian coordinates by x 2 + y 2 = r 2 {\displaystyle x^{2}+y^{2}=r^{2}} , the question is equivalently asking how many pairs of integers m and n there are such that
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.
Given a circle whose center is point O, choose three points V, C, D on the circle. Draw lines VC and VD: angle ∠DVC is an inscribed angle. Now draw line OV and extend it past point O so that it intersects the circle at point E. Angle ∠DVC intercepts arc DC on the circle. Suppose this arc includes point E within it.
Since the distances from that pole point to the tangent points A 1 and B 1 are equal, this pole point must also lie on the radical axis R of the solution circles, by definition (Figure 9). The relationship between pole points and their polar lines is reciprocal; if the pole of L 1 in C 1 lies on R , the pole of R in C 1 must conversely lie on L 1 .