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◦).