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Let q 1 = c 1, let q n = 1 / c n−1, and let q k = (c k − c k−1) / (1 + c k c k−1) for k = 2, ..., n−1. These rational numbers are the tangents of the individual quarter angles, using the formula for the tangent of the difference of angles. Rational side lengths for the polygon circumscribed by the unit circle are thus obtained as s k ...
A circle that is tangent to two sides of a triangle, as the Malfatti circles are, must be centered on one of the angle bisectors of the triangle (green in the figure). These bisectors partition the triangle into three smaller triangles, and Steiner's construction of the Malfatti circles begins by drawing a different triple of circles (shown ...
P ' is the inverse of P with respect to the circle. To invert a number in arithmetic usually means to take its reciprocal. A closely related idea in geometry is that of "inverting" a point. In the plane, the inverse of a point P with respect to a reference circle (Ø) with center O and radius r is a point P ', lying on the ray from O through P ...
A nine-point circle bisects a line segment going from the corresponding triangle's orthocenter to any point on its circumcircle. Figure 4. The center N of the nine-point circle bisects a segment from the orthocenter H to the circumcenter O (making the orthocenter a center of dilation to both circles): [6]: p.152
A circle circumference and radius are proportional. The area enclosed and the square of its radius are proportional. The constants of proportionality are 2 π and π respectively. The circle that is centred at the origin with radius 1 is called the unit circle. Thought of as a great circle of the unit sphere, it becomes the Riemannian circle.
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 pole of a line L in a circle C is a point Q that is the inversion in C of the point P on L that is closest to the center of the circle. Conversely, the polar line (or polar) of a point Q in a circle C is the line L such that its closest point P to the center of the circle is the inversion of Q in C.
The Euler characteristic of a point (and therefore also that of a closed or open disk) is 1. [7] Every continuous map from the closed disk to itself has at least one fixed point (we don't require the map to be bijective or even surjective); this is the case n=2 of the Brouwer fixed-point theorem. [8] The statement is false for the open disk: [9]