= EXAMPLES. 1. Through a given point P draw a line meeting two given lines L and L' divided homographically in corresponding points X, X'. [Join PA, PB, PC, and let these lines meet the axis L' in A", B", C", then ABC... A"B"C"... since the systems are in perspective at P, therefore A"B"C"... = A'B'C'..., and if any point of either coincides with its correspondents of the other, what is required is done; hence lines joining P to the double points of these systems give the two solutions of the problem.] 2. Draw a line through a point P cutting four lines L1, L2, L3, L4 in a row of points A, B, C, D having a given anharmonic ratio k. [Take points A1, A2, A3, on the axis L1, and draw lines cutting the remaining axes in systems of points such that ... The angle LL2 is thus divided homographically by the pairs of rays through C1, D1; C2, D2; C3, D3 ..., etc., and the systems C12C3 D1D2Dз... are therefore equianharmonic.* Join PC1, PC2, PC3, · and let the joining lines meet L in D1, D2, D3', It follows, as in Ex. 1, that D1D2Dз ... = Di'l''Dg'..., and the lines joining their double points to P are those required.] 2 3. Draw a line intersecting five lines such that the anharmonic ratio of any four of the points of intersection is equal to that of any other four. 4. Given two homographic pencils, find the pairs of corresponding rays which intersect on a given line L. [Let the line meet the pencils in points ABC, A'B'C'; the required rays therefore pass through the double points of the homographic rows so determined.] 5. In Ex. 4 find the pair of corresponding rays which intersect at a given angle. [Join the vertices O and O′ of the pencils, and on 00' describe a segment of a circle containing the given angle; let this circle cut the pencils in the points ABC..., A'B'C'..., and find the double points of these homographic systems; therefore, etc.] *This is otherwise evident as all the lines touch the same conic. 6. Find the direction of the parallel rays; and hence draw a transversal to two homographic pencils which shall be divided similarly by them. 7. Find two points on a given line which shall be isogonal conjugates with respect to a given triangle. 8. Construct a triangle with its sides passing through given points and its vertices on given lines, or on a circle. 9. Let the line I joining the vertices of two homographic pencils regarded as a ray of each system have for conjugates L1 and L2 ; prove that any transversal through the point LL2 is cut in involution (cf. Art. 145). 10. Through a given point P draw a line intersecting five lines in the points A, A'; B, B′; P' in any assigned order forming with P an involution. [Let the lines containing A, B meet in O; those containing A', B' in O'. Since (hyp.) O. ABPP'=0'. A'B'P'P=0'. B'A'PP' and the pairs of rays which correspond OB, O'B'; OB, O'A' are fixed; therefore the variable rays OP' and O'P' divide the fifth line L homographically and the double points give the required solutions.] 11. Find a point on a given line such that if joined to five given points any two pairs of connectors shall be in involution with the line and fifth. 12. Describe a circle touching three circles with contacts of assigned species. |