Exercises.
1. Given the altitude of a triangle and the base angles, construct it.
2. From a given point draw to a given line a line making with it an angle equal to a given angle. Show that there will be two solutions.
3. Prove the following construction for trisecting a given line AB:—On AB describe an equilateral △ ABC. Bisect the angles A, B by the lines AD, BD, meeting in D; through D draw parallels to AC, BC, meeting AB in E, F: E, F are the points of trisection of AB.
4. Inscribe a square in a given equilateral triangle, having its base on a given side of the triangle.
5. Draw a line parallel to the base of a triangle so that it may be—1. equal to the intercept it makes on one of the sides from the extremity of the base; 2. equal to the sum of the two intercepts on the sides from the extremities of the base; 3. equal to their difference. Show that there are two solutions in each case.
6. Through two given points in two parallel lines draw two lines forming a lozenge with the given parallels.
7. Between two lines given in position place a line of given length which shall be parallel to a given line. Show that there are two solutions.
PROP. XXXII.—Theorem.
