5. The medians of a triangle divide each other in the ratio of 2 : 1. 6. Construct a triangle, being given two sides and the median of the third side. 7. In every triangle the sum of the medians is less than the perimeter, and greater than three-fourths of the perimeter. 8. Construct a triangle, being given a side and the two medians of the remaining sides. 9. Construct a triangle, being given the three medians. 10. The angle included between the perpendicular from the vertical angle of a triangle on the base, and the bisector of the vertical angle, is equal to half the difference of the base angles. 11. Find in two parallels two points which shall be equidistant from a given point, and whose line of connexion shall be parallel to a given line. 12. Construct a parallelogram, being given two diagonals and a side. 13. The smallest median of a triangle corresponds to the greatest side. 14. Find in two parallels two points subtending a right angle at a given point and equally distant from it. 15. The sum of the distances of any point in the base of an isosceles triangle from the equal sides is equal to the distance of either extremity of the base from the opposite side. 16. The three perpendiculars at the middle points of the sides of a triangle are concurrent. Hence prove that perpendiculars from the vertices on the opposite sides are concurrent [see Ex. 2]. 17. Inscribe a lozenge in a triangle having for an angle one angle of the triangle. 18. Inscribe a square in a triangle having its base on a side of the triangle. 19. Find the locus of a point, the sum or the difference of whose distance from two fixed lines is equal to a given length. 20. The sum of the perpendiculars from any point in the interior of an equilateral triangle is equal to the perpendicular from any vertex on the opposite side. 21. The distance of the foot of the perpendicular from either extremity of the base of a triangle on the bisector of the vertical angle, from the middle point of the base, is equal to half the difference of the sides. 22. In the same case, if the bisector of the external vertical angle be taken, the distance will be equal to half the sum of the sides. 23. Find a point in one of the sides of a triangle such that the sum of the intercepts made by the other sides, on parallels drawn from the same point to these sides, may be equal to a given length. 24. If two angles have their legs respectively parallel, their bisectors are either parallel or perpendicular. 25. If lines be drawn from the extremities of the base of a triangle to the feet of perpendiculars let fall from the same points on either bisector of the vertical angle, these lines meet on the other bisector of the vertical angle. 26. The perpendiculars of a triangle are the bisectors of the angles of the triangle whose vertices are the feet of these perpendiculars.