1. To divide a given line AB internally or externally in the ratio of two given lines, m, n.
Sol.—Through A and B draw any two parallels AC and BD in opposite directions. Cut off AC = m, and BD = n, and join CD; the joining line will divide AB internally at E in the ratio of m : n.
2. If BD′ be drawn in the same direction with AC, as denoted by the dotted line, then CD′ will cut AB externally at E′ in the ratio of m : n.
Cor.—The two points E, E′ divide AB harmonically.
This problem is manifestly equivalent to the following:—Given the sum or difference of two lines and their ratio, to find the lines.
3. Any line AE′, through the middle point B of the base DD′ of a triangle DCD′, is cut harmonically by the sides of the triangle and a parallel to the base through the vertex.
4. Given the sum of the squares on two lines and their ratio; find the lines.
5. Given the difference of the squares on two lines and their ratio; find the lines.
6. Given the base and ratio of the sides of a triangle; construct it when any of the following data is given:—1, the area; 2, the difference on the squares of the sides; 3, the sum of the squares on the sides; 4, the vertical angle; 5, the difference of the base angles.