22. If through the middle point of one side of a triangle a straight line be drawn parallel to the base, it shall bisect the other side; also its length intercepted between the sides of the triangle is half that of the base.

24. The lines which join the middle points of the sides of any quadrilateral figure form a parallelogram: and the perimeter of this parallelogram is equal to the sum of the diagonals of the quadrilateral.

25. ABCD is a parallelogram. E, F the middle points of AB, CD. Shew that BF, DE trisect the diagonal AC.

26. A pavement is to be formed of tiles of the same regular figure. Shew that the only figures which can be used are the equilateral triangle, the square, and the regular hexagon.

27. Shew that admitting two sorts of regular figures of the same sides, the pavement may be formed of

28. In a given line find a point (1) equally distant from two given points, (2) equally distant from two given lines.

29. Find the locus of a point, given

1.

Its distance from a fixed point.

2. Its distance from a fixed line.

3. The sum of its distances from two fixed lines.

4. The difference of its distances from two fixed lines. 30. If two sides of a triangle be produced, the lines which bisect the two exterior angles and the third interior angle meet all in one point.

31. CX, CY are two straight lines at right angles. AB a straight line of fixed length moves so that its extremities are always on CX, CY. Find the locus of the middle point of AB.

CONSTRUCTIONS.

These must not be merely indicated as in Euclid, but drawn accurately to scale with ruler and compass. In this part of the work the assistance of the master is indispensable, and we have therefore offered less explanation of our own. A good many examples of each case should be drawn, using lines and angles of known measurement, and great value should be set on accuracy and finish in the drawing. Many problems in Mechanics and Engineering can be, and practically are, solved by construction alone without calculation. We shall give some examples of this method, which is of sufficient importance to deserve serious attention.

As the circle is necessary to our constructions, we give here the ordinary definition.

A circle is a plane figure contained by one line called the circumference, and is such that all lines drawn from a point within it to the circumference are equal.

This point is called the centre of the circle.

1. To draw a straight line bisecting a given finite straight line at right angles.

Let AB be the given finite straight line.

With centre A and distance greater than half AB describe a circumference; with centre B and same distance describe a

circumference. These circumferences must intersect in two points as at C and D. Join CD meeting AB in E; AB is bisected at right angles in E.

For C and D are both equidistant from A and B; therefore both lie on the line which bisects AB at right angles. [I. 19. Therefore CD is that line.

This is the simplest construction, but it is obviously not essential that the radius of the arcs which intersect below AB, should be the same as that of those which intersect above.

2. To bisect a given angle, as A.

With centre A and any distance describe the arc BC.

With centre B and any distance describe an arc.

With centre C and the same distance describe an arc intersecting the first in D.

A

B

Join AD. AD bisects BAC

3. To draw a line perpendicular to a given line AB.

First. From a given point within it as C.

In CA take any point D; make ▲ CE = CD.

Draw FG, bisecting DE at right angles. This passes through C, and is the line required.

D

bisecting DE at right angles. Draw this line; it shall be the line required.

4. At a given point B in the straight line BC to make an angle equal to the given angle A.

With centre A and any distance AD describe the arc DE. With centre B and the same distance describe the arc CF. Take off with the compass the length DE, and with centre C and this distance describe an arc intersecting CF in F. Join BF.

Then CBF is equal to A.

[I. 13.

For if DE and CF were joined the three sides of the triangle EAD would be equal to those of FBC, each to each.

5. Through a given point to draw a straight line parallel to a given straight line.

6. Through a given point to draw a straight line making a given angle with a given straight line.

7. (1) To construct a triangle having its sides equal to three given straight lines.

Draw a straight line BC equal to the greatest of the three given straight lines. With centre B and radius equal to the second given line describe an arc. With centre C and radius equal to the third given line describe an arc intersecting the