PROBLEMS
1. Draw through a given point a line which shall pass through the inaccessible point of intersection of two given lines. The following construction may be made to depend upon Desargues's theorem: Through the given point P draw any two rays cutting the two lines in the points AB' and A'B, A, B, lying on one of the given lines and A', B', on the other. Join AA' and BB', and find their point of intersection S. Through S draw any other ray, cutting the given lines in CC'. Join BC' and B'C, and obtain their point of intersection Q. PQ is the desired line. Justify this construction.
2. To draw through a given point P a line which shall meet two given lines in points A and B, equally distant from P. Justify the following construction: Join P to the point S of intersection of the two given lines. Construct the fourth harmonic of PS with respect to the two given lines. Draw through P a line parallel to this line. This is the required line.
3. Given a parallelogram in the same plane with a given segment AC, to construct linearly the middle point of AC.
4. Given four harmonic lines, of which one pair are at right angles to each other, show that the other pair make equal angles with them. This is a theorem of which frequent use will be made.
5. Given the middle point of a line segment, to draw a line parallel to the segment and passing through a given point.
6. A line is drawn cutting the sides of a triangle ABC in the points A', B', C' the point A' lying on the side BC, etc. The harmonic conjugate of A' with respect to B and C is then constructed and called A". Similarly, B" and C" are constructed. Show that A"B"C" lie on a straight line. Find other sets of three points on a line in the figure. Find also sets of three lines through a point.