56 VII. CONGRUENT LINES IN A TRIANGLE Figure 53 Exercises Exercise 33. Join a given point to the intersection of two lines, which intersect outside the limits of the diagram (53). Exercise 34. In a trapezoid, the midpoints of the non-parallel sides and the midpoints of the two diagonals are on the same line, parallel to the bases. The distance between the midpoints of the non-parallel sides equals half the sum of the bases the distance between the midpoints of the diagonals is equal to half their difference. Exercise 35. If, from two points A, B and the midpoint C of AB, we drop perpendiculars onto an arbitrary line, the perpendicular from C is equal to half the sum of the other two perpendiculars, or to half their difference, according as whether these two perpendiculars have the same or opposite sense. Exercise 36. The midpoints of the sides of any quadrilateral are the vertices of a parallelogram. The sides of this parallelogram are parallel to the diagonals of the given quadrilateral, and equal to halves of these diagonals. The center of the parallelogram is also the midpoint of the segment joining the midpoints of the diagonals of the given quadrilateral. Exercise 37. Prove that the medians of a triangle ABC are concurrent by extending the median CF (Fig. 53) beyond F by a length equal to F G. Exercise 38. Given three lines passing through the same point O (all three distinct), and a point A on one of them, show that there exists: 1◦. A triangle with a vertex at A and having the three lines as its altitudes (one exception) 2◦. A triangle with a vertex at A and having the three lines as its medians 3◦. A triangle with a vertex at A and having the three lines as bisectors of its interior or exterior angles (one exception) 4◦. A triangle with a midpoint of one of its sides at point A and having the three lines as perpendicular bisectors of the sides (reduce this to 1◦).

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