23. How many figures similar to a given rectilineal figure of sides can be described on a given line? 24. How many centres of similitude can two regular polygons of n sides each have? Ans. n centres, which lie on a circle. 25. What are homothetic figures? 26. How do the areas of similar rectilineal figures vary? 27. What proposition is xix. a special case of? 28. Define Philo’s line. 29. How many centres of similitude have two circles? Exercises on Book VI. 1. If in a fixed triangle we draw a variable parallel to the base, the locus of the points of intersection of the diagonals of the trapezium thus cut off from the triangle is the median that bisects the base. 2. Find the locus of the point which divides in a given ratio the several lines drawn from a given point to the circumference of a given circle. 3. Two lines AB, XY , are given in position: AB is divided in C in the ratio m : n, and parallels AA′, BB′, CC′, are drawn in any direction meeting XY in the points A′, B′, C′; prove 4. Three concurrent lines from the vertices of a triangle ABC meet the opposite sides in A′, B′, C′; prove 5. If a transversal meet the sides of a triangle ABC in the points A′, B′, C′; prove 6. If on a variable line AC, drawn from a fixed point A to any point B in the circumference of a given circle, a point C be taken such that the rectangle AB.AC is constant, the locus of C is a circle. 7. If D be the middle point of the base BC of a triangle ABC, E the foot of the perpendicular, L the point where the bisector of the angle A meets BC, H the point of contact of the inscribed circle with BC; prove DE.HL = HE.HD. 8. In the same case, if K be the point of contact with BC of the escribed circle, which touches the other sides produced, LH.BK = BD.LE. 9. If R, r, r′, r′′, r′′′ be the radii of the circumscribed, the inscribed, and the escribed circles of a plane triangle, d, d′, d′′, d′′′ the distances of the centre of the circumscribed circle from the centres of the others, then R2 = d2 + 2Rr = d′2 − 2Rr′, &c. 10. In the same case, 12R2 = d2 + d′2 + d′′2 + d′′′2. 11. If p′, p′′, p′′′ denote the perpendiculars of a triangle, then (1) + + = ; (2) + − = , &c.; (3) = −, &c.; (4) = + , &c. 12. In a given triangle inscribe another of given form, and having one of its angles at a given point in one of the sides of the original triangle. 13. If a triangle of given form move so that its three sides pass through three fixed points, the locus of any point in its plane is a circle. 14. The angle A and the area of a triangle ABC are given in magnitude: if the point A be fixed in position, and the point B move along a fixed line or circle, the locus of the point C is a circle.