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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.
one solves the line equation for x or y and substitutes it into the equation of the circle and gets for the solution (using the formula of a quadratic equation) (,), (,) with x 1 / 2 = a c ± b r 2 ( a 2 + b 2 ) − c 2 a 2 + b 2 , {\displaystyle x_{1/2}={\frac {ac\pm b{\sqrt {r^{2}(a^{2}+b^{2})-c^{2}}}}{a^{2}+b^{2}}}\ ,}
This formula cannot be used if the quadrilateral is a right kite, since the denominator is zero in that case. If M, N are the midpoints of the diagonals, and E, F are the intersection points of the extensions of opposite sides, then the area of a bicentric quadrilateral is given by
Draw a circle centered at M through the point A. This is the Carlyle circle for x 2 + x − 1 = 0. Mark its intersection with the horizontal line (inside the original circle) as the point W and its intersection outside the circle as the point V. These are the points p 1 and p 2 mentioned above. Draw a circle of radius OA and center W. It ...
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).
Using this form, vertical lines correspond to equations with b = 0. One can further suppose either c = 1 or c = 0, by dividing everything by c if it is not zero. There are many variant ways to write the equation of a line which can all be converted from one to another by algebraic manipulation. The above form is sometimes called the standard form.
It is relatively straightforward to construct a line t tangent to a circle at a point T on the circumference of the circle: A line a is drawn from O, the center of the circle, through the radial point T; The line t is the perpendicular line to a. Construction of a tangent to a given circle (black) from a given exterior point (P).
The area of a regular polygon is half its perimeter multiplied by the distance from its center to its sides, and because the sequence tends to a circle, the corresponding formula–that the area is half the circumference times the radius–namely, A = 1 / 2 × 2πr × r, holds for a circle.