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.