Search results
Results from the WOW.Com Content Network
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.
Specifically, draw a diagonal line connecting two points on the diagram so that every other point is either on or to the right and above it. There is at least one such line if the curve passes through the origin. Let the equation of the line be qα+pβ=r. Suppose the curve is approximated by y=Cx p/q near the origin.
Given a point Z not on conic C, draw two secants from Z through C crossing at points A, B, D, and E. Then these four points form a complete quadrangle, and Z is at one of the diagonal points. The line joining the other two diagonal points is the polar of Z, and Z is the pole of this line. [2]
One diagonal crosses the midpoint of the other diagonal at a right angle, forming its perpendicular bisector. [9] (In the concave case, the line through one of the diagonals bisects the other.) One diagonal is a line of symmetry. It divides the quadrilateral into two congruent triangles that are mirror images of each other. [7]
A set of sides that can form a cyclic quadrilateral can be arranged in any of three distinct sequences each of which can form a cyclic quadrilateral of the same area in the same circumcircle (the areas being the same according to Brahmagupta's area formula). Any two of these cyclic quadrilaterals have one diagonal length in common. [17]: p. 84
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 ...
Any line through a triangle that splits both the triangle's area and its perimeter in half goes through the triangle's incenter (the center of its incircle). There are either one, two, or three of these for any given triangle. A line through the incenter bisects one of the area or perimeter if and only if it also bisects the other. [12]
In geometry, an intersection is a point, line, or curve common to two or more objects (such as lines, curves, planes, and surfaces). The simplest case in Euclidean geometry is the line–line intersection between two distinct lines, which either is one point (sometimes called a vertex) or does not exist (if the lines are parallel). Other types ...