Constructible Number - Geometric Definitions

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

1. Z is in the intersection of L(E, F ) and L(G, H ), where L(E, F ) ≠ L(G, H );
2. Z is in the intersection of C (E, F ) and C (G, H ), where C (E, F ) ≠ C (G, H );
3. 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

1. Z = A;
2. Z = A′;
3. there exist points P₁, ..., P_n, with Z = P_n, such that for all j ≥ 1, P_{j + 1} is constructible from points in the set {A, A′, P₁, ..., 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′).