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The set of points equidistant from two points is a perpendicular bisector to the line segment connecting the two points. [8] The set of points equidistant from two intersecting lines is the union of their two angle bisectors. All conic sections are loci: [9] Circle: the set of points at constant distance (the radius) from a fixed point (the ...
O. Radko and E. Tsukerman, The Perpendicular Bisector Construction, the Isoptic Point and the Simson Line of a Quadrilateral, Forum Geometricorum 12: 161–189 (2012).
Concurrent perpendicular bisectors of chords between concyclic points Circumscribed circle, C, and circumcenter, O, of a cyclic polygon, P. In geometry, a set of points are said to be concyclic (or cocyclic) if they lie on a common circle.
The interior perpendicular bisector of a side of a triangle is the segment, falling entirely on and inside the triangle, of the line that perpendicularly bisects that side. The three perpendicular bisectors of a triangle's three sides intersect at the circumcenter (the center of the circle through the three vertices). Thus any line through a ...
Perpendicular bisector construction of a quadrilateral, on the use of perpendicular bisectors of a quadrilateral's sides to form another quadrilateral Topics referred to by the same term This disambiguation page lists articles associated with the title Perpendicular bisector construction .
Twelve key lengths of a triangle are the three side lengths, the three altitudes, the three medians, and the three angle bisectors. Together with the three angles, these give 95 distinct combinations, 63 of which give rise to a constructible triangle, 30 of which do not, and two of which are underdefined. [13]: pp. 201–203
The three perpendicular bisectors meet at the circumcenter. Other sets of lines associated with a triangle are concurrent as well. For example: Any median (which is necessarily a bisector of the triangle's area) is concurrent with two other area bisectors each of which is parallel to a side. [1]
For one other site , the points that are closer to than to , or equally distant, form a closed half-space, whose boundary is the perpendicular bisector of line segment . Cell R k {\displaystyle R_{k}} is the intersection of all of these n − 1 {\displaystyle n-1} half-spaces, and hence it is a convex polygon . [ 6 ]