- Centre (geometry)
geometry, the centre (or center, in American English) of an object is a point in some sense in the middle of the object. If geometry is regarded as the study of isometry groups then the centre is a fixed point of the isometries.
The centre of a
circleis the point equidistant from the points on the edge. Similarly the centre of a sphereis the point equidistant from the points on the surface, and the centre of a line segment is the midpoint of the two ends.
For objects with several
symmetries, the centre of symmetry is the point left unchanged by the symmetric actions. So the centre of a square, rectangle, rhombusor parallelogramis where the diagonals intersect, this being (amongst other properties) the fixed point of rotational symmetries. Similarly the centre of an ellipseis where the axes intersect.
Several special points of a triangle are often described as centres: the
circumcentre, centroidor centre of mass, incentre, excentres, orthocentre, nine-point centre. For an equilateral triangle, these (except for the excentres) are the same point.
A strict definition of a triangle centre is a point whose
trilinear coordinates are "f"("a","b","c") : "f"("b","c","a") : "f"("c","a","b") where "f" is a function of the lengths of the three sides of the triangle, "a", "b", "c" such that:
# "f" is homogenous in "a", "b", "c" i.e. "f"("ta","tb","tc")="t""h""f"("a","b","c") for some real power "h"; thus the position of a centre is independent of scale.
# "f" is symmetric in its last two arguments i.e. "f"("a","b","c")= "f"("a","c","b"); thus position of a centre in a mirror-image triangle is the mirror-image of its position in the original triangle. [ [http://faculty.evansville.edu/ck6/tcenters/roads.html Algebraic Highways in Triangle Geometry] ]
This strict definition exclude the excentres, and also excludes pairs of bicentric points such as the
Brocard points (which are interchanged by a mirror-image reflection). The Encyclopedia of Triangle Centerslists over 3,000 different triangle centres.
Fixed points of isometry groups in Euclidean space
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