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In the other direction, a set of geometric objects may be specified by algebraically constructible real numbers: coordinates for points, slope and -intercept for lines, and center and radius for circles. It is possible (but tedious) to develop formulas in terms of these values, using only arithmetic and square roots, for each additional object ...
the nine-point centre, the centre of the circle that passes through nine key points of the triangle. For an equilateral triangle , these are the same point, which lies at the intersection of the three axes of symmetry of the triangle, one third of the distance from its base to its apex.
The equation of a circle is (x − a) 2 + (y − b) 2 = r 2 where a and b are the coordinates of the center (a, b) and r is the radius. Cartesian coordinates are named for René Descartes, whose invention of them in the 17th century revolutionized mathematics by allowing the expression of problems of geometry in terms of algebra and calculus.
The distance between any point of the circle and the centre is called the radius. The length of a line segment connecting two points on the circle and passing through the centre is called the diameter. A circle bounds a region of the plane called a disc. The circle has been known since before the beginning of recorded history.
Consider a circle P with center O and a point A which may lie inside or outside the circle P. Take the intersection point C of the ray OA with the circle P. Connect the point C with an arbitrary point B on the circle P (different from C and from the point on P antipodal to C) Let h be the reflection of ray BA in line BC. Then h cuts ray OC in a ...
Angle AOB is a central angle. A central angle is an angle whose apex (vertex) is the center O of a circle and whose legs (sides) are radii intersecting the circle in two distinct points A and B. Central angles are subtended by an arc between those two points, and the arc length is the central angle of a circle of radius one (measured in radians). [1]
Thales's theorem can be used to construct the tangent to a given circle that passes through a given point. In the figure at right, given circle k with centre O and the point P outside k, bisect OP at H and draw the circle of radius OH with centre H. OP is a diameter of this circle, so the triangles connecting OP to the points T and T′ where ...
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).