**Geometric Definitions**

The geometric definition of a constructible point is as follows. First, for any two distinct points *P* and *Q* in the plane, let *L*(*P*, *Q* ) denote the unique line through *P* and *Q*, and let *C* (*P*, *Q* ) denote the unique circle with center *P*, passing through *Q*. (Note that the order of *P* and *Q* matters for the circle.) By convention, *L*(*P*, *P* ) = *C* (*P*, *P* ) = {*P* }. Then a point *Z* is *constructible from E, F, G and H* if either

*Z*is in the intersection of*L*(*E*,*F*) and*L*(*G*,*H*), where*L*(*E*,*F*) ≠*L*(*G*,*H*);*Z*is in the intersection of*C*(*E*,*F*) and*C*(*G*,*H*), where*C*(*E*,*F*) ≠*C*(*G*,*H*);*Z*is in the intersection of*L*(*E*,*F*) and*C*(*G*,*H*).

Since the order of *E*, *F*, *G*, and *H* in the above definition is irrelevant, the four letters may be permuted in any way. Put simply, *Z* is constructible from *E*, *F*, *G* and *H* if it lies in the intersection of any two distinct lines, or of any two distinct circles, or of a line and a circle, where these lines and/or circles can be determined by *E*, *F*, *G*, and *H*, in the above sense.

Now, let *A* and *A*′ be any two distinct fixed points in the plane. A point *Z* is *constructible* if either

*Z*=*A*;*Z*=*A*′;- there exist points
*P*_{1}, ...,*P*_{n}, with*Z*=*P*_{n}, such that for all*j*≥ 1,*P*_{j + 1}is constructible from points in the set {*A*,*A*′,*P*_{1}, ...,*P*_{j}}.

Put simply, *Z* is constructible if it is either *A* or *A*′, or if it is obtainable from a finite sequence of points starting with *A* and *A*′, where each new point is constructible from previous points in the sequence.

For example, the center point of *A* and *A*′ is defined as follows. The circles *C* (*A*, *A*′) and *C* (*A*′, *A*) intersect in two distinct points; these points determine a unique line, and the center is defined to be the intersection of this line with *L*(*A*, *A*′).

Read more about this topic: Constructible Number

### Famous quotes containing the words definitions and/or geometric:

“What I do not like about our *definitions* of genius is that there is in them nothing of the day of judgment, nothing of resounding through eternity and nothing of the footsteps of the Almighty.”

—G.C. (Georg Christoph)

“In mathematics he was greater

Than Tycho Brahe, or Erra Pater:

For he, by *geometric* scale,

Could take the size of pots of ale;

Resolve, by sines and tangents straight,

If bread and butter wanted weight;

And wisely tell what hour o’ th’ day

The clock doth strike, by algebra.”

—Samuel Butler (1612–1680)