FIG. 49.

To bisect a triangle A B C by a line parallel to its perpendicular altitude.

Bisect the base A B in D.

From C the apex of the triangle drop C E perpendicular to the base.

Find a mean proportional A F between the unequal segments А Е, Е В.

Mark off from A H to A F, and draw line parallel to C E. The triangle A B C will be bisected as required.

PROBLEM 50.

To divide a triangle A B C into any number of equal parts by lines drawn from the apex.

By rule given, we have that when triangles have the same altitude and equal bases they are equal to one another. Therefore, if we divide the base into the required number of equal parts, and draw lines from the points of division to the apex, the triangle will be divided into the required number of equal parts.

FIG. 51.

To divide the triangle A B C into any number of equal parts by lines drawn from a given point P in one side.

Divide the side A C into the number of parts the triangle requires dividing into (say three-1, 2, 3).

Join P B.

From 1, 2, draw 1 1', 2 2', parallel to P B.

Join P 2' and P 1', and the triangle will be divided as required.

PROBLEM 52.

To divide a square into any number of equal parts by lines parallel to the diagonal.

This problem is similar to Fig. 48. The divisions have to be made upon one side, and then proceed as already indicated, the divisions being marked off at equal distances from the diagonal.

PROBLEM 53.

To divide a square into any number of equal parts by a line drawn from one corner.

Divide the two opposite adjacent sides into as many equal parts as the square is required in, half the divisions to be on each side. Join the points of division and the opposite corner with the corner, and the triangle will be divided as required.

CHAPTER V.

CONSTRUCTIVE PROBLEMS ON PREVIOUS CHAPTERS.

FIG. 54.

To inscribe a square within any triangle A B C.

Through C draw C E parallel to A B, making C E equal to the altitude of the triangle C K.

Join B E, cutting A C in the point D.

From D drop a perpendicular D G to the base.

On D G construct a square, which will be what is required.

FIG. 55.

To inscribe a circle within any triangle.

Bisect any two of the angles, and produce the line of bisection until they intersect at D.

The point D is the centre of the triangle.

From D drop a perpendicular to one of the sides.

With centre D and radius equal to perpendicular describe circle, which will be inscribed in the given triangle.

FIG. 56.

To describe a circle about any triangle.

Bisect two of the adjacent sides, and produce the line of bisection until they meet in D.

Join D and one of the angular points C.

With centre D and radius DC describe a circle, which will answer the required conditions.

FIG. 57.

To describe a triangle about a circle, such triangle to have angles equal to those of a given triangle A B C.

Produce the base of the given triangle A B C.

Draw the radii D E, D F, and D G, making angles with each other equal to the exterior angles of the triangle A B C.