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The number of points (n), chords (c) and regions (r G) for first 6 terms of Moser's circle problem. In geometry, the problem of dividing a circle into areas by means of an inscribed polygon with n sides in such a way as to maximise the number of areas created by the edges and diagonals, sometimes called Moser's circle problem (named after Leo Moser), has a solution by an inductive method.
Circle with square and octagon inscribed, showing area gap. Suppose that the area C enclosed by the circle is greater than the area T = cr/2 of the triangle. Let E denote the excess amount. Inscribe a square in the circle, so that its four corners lie on the circle. Between the square and the circle are four segments.
Since cos(180° − θ) = −cos θ, we have cos 2 (180° − θ) = cos 2 θ.) This more general formula is known as Bretschneider's formula. It is a property of cyclic quadrilaterals (and ultimately of inscribed angles) that opposite angles of a quadrilateral sum to 180°.
If a tangent from an external point A meets the circle at F and a secant from the external point A meets the circle at C and D respectively, then AF 2 = AC × AD (tangent–secant theorem). The angle between a chord and the tangent at one of its endpoints is equal to one half the angle subtended at the centre of the circle, on the opposite side ...
A circle with an equilateral chord (red). One sixtieth of this arc is a degree. Six such chords complete the circle. [6]The original motivation for choosing the degree as a unit of rotations and angles is unknown.
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 ...
The nine-point circle is tangent to the incircle and excircles. In geometry, the nine-point circle is a circle that can be constructed for any given triangle. It is so named because it passes through nine significant concyclic points defined from the triangle. These nine points are: [28] [29] The midpoint of each side of the triangle; The foot ...
The case H = 0 can be eliminated as degenerate, so the tangential equation of C can be written as P + fQ = 0 where f is an arbitrary polynomial of degree 2m. [1] For example, let m = 2, P 1 = (1, 0), and P 2 = (−1, 0). The tangential equations are + = = so P = X 2 − 1 = 0. The tangential equations for the circular points at infinity are