Homework Assignment 3
pg.50 prob. 3
Suppose that A,B, and C are noncollinear points in an affine plane. Prove that there is a unique point D such that (A,B,C,D) is a parallelogram.
Construct the line l joining A and B, and the line m joining B and C. Since A,B and C are noncollinear, the lines l and m are distinct. Since C is not on l, we can construct the unique line k, through C and parallel to l. Since A is not on m, we can construct the unique line n, through A and parallel to m. Since n intersects one of the parallel lines l and k, it must intersect the other. Let the point of intersection of n and k be D. (A,B,C,D) is clearly a parallelogram. Now suppose that (A,B,C,D') is a parallelogram. Since the line joining A and D' is parallel to the line joining B and C (i.e. m), it must be n, since there is a unique line through A parallel to m. Thus, D' is on n. Similarly, we can show that D' must be on k. Hence, D = D' since n and k can intersect in only one point.
Make sketches of the following configurations:
- (A,B,C,D) is a quadrilateral with no pair of its sides parallel.
- In the configuration from (a) insert A' so that (A',B,C,D) is a parallelogram.
- In the configuration from (b) insert B' so that (A,B',C,D) is a parallelogram.
- Similarly, insert C' and D'.
Complete the addition table for (OIDF). What is the inverse of D?
The additive inverse of D is D.
Suppose that (P,L) is a Desarguesian affine plane in which the diagonals of every parallelogram are parallel. Suppose that A is any point in P. Prove that relative to any O, A+A = O.
Let A and O be any two points in the Desarguesian affine plane. If A = O, the statement is true, so we may assume that these points are distinct. Let l be the line determined by A and O. Choose B, any point not on l and draw the line m through B parallel to l. Let k be the line joining O and B and n the line through A parallel to k. The lines l and n meet at a point D. Notice that (BOAD) is a parallelogram. To determine A+A, we need to construct the line parallel to the line determined by A and B which passes through D. Since the line determined by A and B is a diagonal of the parallelogram, the unique line through D which is parallel to it is the other diagonal (by hypothesis), i.e. the line joining O and D. Since this line meets l at O, we have A+A = O.
Let (A,B,C,D) be a quadrilateral in a Desarguesian affine plane, none of whose sides are parallel. Define A' to be the unique point such that (A',B,C,D) is a parallelogram, and B' the unique point such that (A,B',C,D) is a parallelogram. Prove that the line joining A and A' is parallel to the line joining B and B'.
By construction, lines l(C,D) and l(A,B') are parallel, as are the lines l(C,D) and l(A',B). Consider the triangles DAA' and CB'B. The lines joining corresponding vertices of these triangles are mutually parallel. Also by construction, lines l(A,D) and l(C,B') are parallel, as are l(D,A') and l(C,B). By Desargues theorem, the third sides of these triangles, namely l(A,A') and l(B',B) are parallel.
Let O, I and A be distinct collinear points in a Desarguesian affine plane. Define the point 2I to be I + I and 2A to be the point A + A, the addition done relative to O. Prove geometrically that (2I)A = A + A assuming that the points O, I, 2I, A and 2A are all distinct.
Choose point B and construct 2I.
Using B, construct (2I)A.
Again using B, construct 2A.
To prove that (2I)A = A + A = 2A, we must show that E = E'. Consider triangles DD'E' and IBA. Since, by construction, l(B,A) || l(D',E'), l(I,B) || l(D,D') and l(D,E') || l(I,A) (i.e., the corresponding sides of the triangles are parallel in pairs) and l(D',B) || l(I,D) we have by the converse of Desargues theorem (affine version) that l(A,E') || l(I,D || l(O,B). Thus, l(A,E') = l(A,E) and the conclusion follows.