Radical Axis and Central Line
Except for the two special cases of a pencil of concentric circles and a pencil of coincident lines, any two circles within a pencil have the same radical axis, and all circles in the pencil have collinear centers. Any three or more circles from the same family are called coaxal circles or coaxial circles.
The elliptic pencil of circles passing through the two points C and D (the set of red circles, in the figure) has the line CD as its radical axis. The centers of the circles in this pencil lie on the perpendicular bisector of CD. The hyperbolic pencil defined by points C and D (the blue circles) has its radical axis on the perpendicular bisector of line CD, and all its circle centers on line CD.
The radical axis of any pencil of circles, interpreted as an infinite-radius circle, belongs to the pencil. Any three circles belong to a common pencil whenever all three pairs share the same radical axis and their centers are collinear.
Read more about this topic: Apollonian Circles
Famous quotes containing the words radical, axis, central and/or line:
“The most radical revolutionary will become a conservative the day after the revolution.”
—Hannah Arendt (1906–1975)
“A book is not an autonomous entity: it is a relation, an axis of innumerable relations. One literature differs from another, be it earlier or later, not because of the texts but because of the way they are read: if I could read any page from the present time—this one, for instance—as it will be read in the year 2000, I would know what the literature of the year 2000 would be like.”
—Jorge Luis Borges (1899–1986)
“There is no such thing as a free lunch.”
—Anonymous.
An axiom from economics popular in the 1960s, the words have no known source, though have been dated to the 1840s, when they were used in saloons where snacks were offered to customers. Ascribed to an Italian immigrant outside Grand Central Station, New York, in Alistair Cooke’s America (epilogue, 1973)
“Their bodies are buried in peace; but their name liveth for evermore.”
—Apocrypha. Ecclesiasticus, 44:14.
The line “their name liveth for evermore” was chosen by Rudyard Kipling on behalf of the Imperial War Graves Commission as an epitaph to be used in Commonwealth War Cemeteries. Kipling had himself lost a son in the fighting.