What Is Mathematics? An Elementary Approach to Ideas and Methods, 2nd Edition (1996)
CHAPTER IV. PROJECTIVE GEOMETRY. AXIOMATICS. NON-EUCLIDEAN GEOMETRIES
§7. PROBLEMS ON CONSTRUCTIONS WITH THE STRAIGHT-EDGE ALONE
In the constructions helow it is understood that only the straightedge is admitted as tool.
Problems 1 to 18 are contained in a paper by J. Steiner in which he proves that the compass can be dispensed with as a tool for geometrical constructions if a fixed circle with its center is given (see Chapt. III, p. 151). The reader is advised to solve these problems in the order given.
A set of four lines a, b, c, d through a point P is called harmonic, if the cross-ratio (abcd) equals –1. a and b are said to be conjugate with respect to c and d, and vice versa.
1) Prove: If, in a set of four harmonic lines a, b, c, d, the ray a bisects the angle between c and d, then b is perpendicular to a.
2) Construct the fourth harmonic line to three given lines through a point. (Hint: Use the theorem on the complete quadrilateral.)
3) Construct the fourth harmonic point to three points on a line.
4) If a given right angle and a given arbitrary angle have their vertex and one side in common, double the given arbitrary angle.
5) Given an angle and its bisector b. Construct a perpendicular to b through the vertex P of the given angle.
6) Prove: If the lines l1, l2, l3,..., ln through a point P intersect the straight line a in the points A1, A2,...,An and intersect the line b in the points B1, B2,..., Bn, then all the intersections of the pairs of lines AiBk and AkBi (i ≠ k; i, k = 1, 2,..., n) lie on a straight line.
7) Prove: If a parallel to the side BC of the triangle ABC intersects AB in B′ and AC in C′, then the line joining A with the intersection D of B ′C and C ′B bisects BC.
7a) Formulate and prove the converse of 7.
8) On a straight line l three point P, Q, R are given, such that Q is the midpoint of the segment PR. Construct a parallel to l through a given point S.
9) Given two parallel lines l1 and l2; bisect a given segment AB on l1.
10) Draw a parallel through a given point P to two given parallel lines l1 and l2. (Hint: Reduce 9 to 7 using 8.)
11) Steiner gives the following solution to the problem of doubling a given line segment AB when a parallel l to AB is given: Through a point C not on l nor on the line AB draw CA intersecting l at A1, CB intersecting l at B1. Then (see 10) draw a parallel to l through C, which meets BA1at D. If DB1 meets AB at E, then AE = 2·AB.
Prove the last statement.
12) Divide a segment AB into n equal parts if a parallel l to AB is given. (Hint: Construct first the n-fold of an arbitrary segment on l, using 11.)
13) Given a parallelogram ABCD, draw a parallel through a point P to a straight line l. (Hint: Apply 10 to the center of the parallelogram and use 8.)
14) Given a parallelogram, multiply a given segment by n. (Hint: Use 13 and 11.)
15) Given a parallelogram, divide a given segment into n parts.
16) If a fixed circle and its center are given, draw a parallel to a given straight line through a given point. (Hint: Use 13.)
17) If a fixed circle and its center are given, multiply and divide a given segment by n. (Hint: Use 13.)
18) Given a fixed circle and its center, draw a perpendicular to a given line through a given point. (Hint: Using a rectangle inscribed in the fixed circle and having two sides parallel to the given line, reduce to previous exercised.)
19) Using the results of problems 1–18, which basic construction problems can you solve if your tool is a ruler with two parallel edges?
20) Two given straight lines l1 and l2 intersect at a point P outside the given sheet of paper. Construct the line joining a given point Q with P. (Hint: Complete the given elements to the figure of Desargues’s theorem for the plane in such a way that P and Q become intersections of corresponding sides of the two triangles in Desargues’s Theorem.)
21) Construct the line joining two given points whose distance is greater than the length of the straightedge used. (Hint: Use 20.)
22) Two points P and Q outside the given sheet of paper are determined by two pairs of straight lines l1, l2 and m1, m2 through P and Q, respectively. Construct that part of the line PQ that lies on the given sheet of paper. (Hint: To obtain a point of PQ complete the given elements to a figure of Desargues’s theorem in such a way that one triangle has two sides on l1 and m1 and the other one corresponding sides on l2 and m2.)
23) Solve 20 by means of Pascal’s theorem (p. 188). (Hint: Complete the given elements to a figure of Pascal’s theorem, using l1, l2 as a pair of opposite sides of the hexagon and Q as point of intersection of another pair of opposite sides.)
*24) Two straight lines entirely outside the given sheet of paper are each given by two pairs of straight lines intersecting at points of the lines outside the paper. Determine their point of intersection by a pair of lines through it.