33. Let AB be a diameter of a circle, and C any point on the circumference; draw the straight lines AC and BC: then the angle ACB will be a right angle.

34.

Let ABC and DEF be two triangles such that the angle A is equal to the angle D, the angle B to the

F

AA

angle E, and the angle C to the angle F: then the sides opposite the equal angles will be proportionals.

That is, if EF be double of BC, then FD is double of CA, and DE is double of AB; if EF be three times BC, then FD is three times CA, and DE is three times AB; and so on. The two triangles are said to be similar.

35. Let ABC and DEF be similar triangles, the angles C and F being corresponding angles; let CG and FH

A A

A G

be perpendiculars from C and F on the opposite sides: then CG will be to AB as FH is to DE.

38. Let AB and CD be two chords of a circle; let them meet, produced if necessary, at E; join BC and

E

B

AD: then the triangles AED and BEC will be similar, the angles EAD and ECB being equal, and the angles EDA and EBC being equal.

See Art. 32.

III. PROBLEMS.

39. We shall now give the solutions of a few problems which occur in practice when it is necessary to draw figures accurately. We suppose that a ruler and compasses are employed; these instruments will be sufficient for our purpose. Other instruments are often useful, such as a square and parallel rulers; but they are not absolutely necessary.

The solutions which we shall give of the problems depend mainly on the principles stated in Chapter II; there will not be much difficulty in verifying practically the correctness of the results, and those who make themselves acquainted with the elements of demonstrative geometry will perceive the rigorous exactness of the processes.

40. To divide a given straight line into two equal parts.

Let AB be the given straight

line. From the centres A and B with a radius greater than half AB describe arcs cutting each other at D and E. Join DE, cutting AB at C. Then AC will be equal to CB. The straight line DE will be at right angles to AB, so that we see how to draw a straight line which shall be at right angles to a given straight line and shall also divide it into two equal parts.

A

B

41. To divide a given angle into two equal parts.

Let ABC be the given angle. From the centre B with any radius describe an arc, cutting BA at D, and BC at E. From D and E as centres with any sufficient radius describe arcs cutting each other at F. Join BF. Then the angle ABF will be equal to the angle CBF.

B

D

E

42. To draw a straight line parallel to a given straight line, and at a given distance from it.

Let AB be the given straight line, and let C be equal to the given distance. From any two points D and E in

AB as centres, with a radius equal to C, describe arcs. Draw a straight line FG touching these arcs. Then FG will be parallel to AB, and at a distance from it equal to C.

43. To make a triangle having its sides equal to three given straight lines.

Let A, B, and C be the given straight lines.

Draw a straight line DE equal to one of the given straight lines, A. From the centre D, with a radius equal to B, describe an arc; and from the centre E with a radius equal to C describe another arc. Let these arcs cut each other at F. Join DF and EF. Then DEF will be the triangle required.

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

Let A be the given point, and BCthe given straight line. Take any point D in BC. From the centre D, with the radius DA describe an arc, cutting BC at E, and draw the chord AE. B

From the centre A, with

E

the radius AD, describe an arc, and draw the chord DF equal to the chord AE. Join AF. Then AF will be parallel to BC.

T. M.