§ 304. DEFINITION.-If two similar rectilinear figures are placed so that every straight line that joins two corresponding vertices passes through a certain point, then this point is called the centre of similitude of the figures.

THEOREM [Pn.6]. Two similar polygons can always be

placed so as to have a centre of similitude.

ABCDE, abcde, are two similar polygons, in which the vertices A, B, C, . . . correspond respectively to the vertices a, b, c,

E

.

Cons. Place the two polygons so that corresponding sides AB and ab are parallel; and so that BC and be are on the same sides of AB and ab respectively, if AB and ab are drawn in the same sense (Fig. 439); or so that BC and be are on opposite sides of AB and ab respectively, if AB and ab are drawn in opposite senses (Fig. 440).

Let Aa and Bb meet in 0, and let OC meet bc at c'.

Proof. In either figure ABC LABO = Lab0;

Again,

=

Labc (Hyp.), and

P.2c. or P.1c.

▲ Obc, whence BC || bc. P.2. or P.1.

AB || ab and BC || be',

A OABA Oab and A OBC ||| Obc';

S.1.

hence c' coincides with c; that is to say, OC passes

through c.

Similarly it can now be proved that CD || cd, and OD passes through d; and so on.

Hence O is a centre of similitude for the polygons

ABCDE, abcde.

305. On the Theory of Similar Figures. In the preceding portion of this chapter we have dealt with similar triangles and similar polygons. To complete the theory of similar figures we must include any two similar aggregations of points, whether the points are isolated or form continuous lines, straight or curved. This would include, for example, the case of two maps of the same country drawn to different scales (see Fig. 441).

DEFINITION. Two geometrical figures are similar, if to any point on the one figure there corresponds one point on the other, and if the ratio of the join of any two points on the one figure to the join of the corresponding two points on the other figure is constant. This constant ratio is called the ratio of similitude.

0

For example, in Fig. 441 the points A, B, C, D, E. on the one map correspond respectively to the points a, b, c, d, e, on the other; also

Thus these two maps are similar, and the ratio of similitude is 1.7.

In the same way each of the two larger maps is similar to the smallest map.

The important propositions in the theory of these similar figures are given in the following set of riders.

EXERCISES CXV (Riders).

1. [SF.1]. It is always possible to construct a figure similar to a given figure with a given ratio of similitude k.

be various points on the given figure (Fig. 441). On Oa, Ob, Oc, take points A, B, C, ос

Let a, b, c, d, Take any point 0. ОА OB

such that

=

=

...

=

k.

AC BE HD

=

ac be hd

DEFINITION. In similar figures corresponding lines are paths traced by corresponding points.

B

2. [SF.2]. -In similar figures the join of any two points corresponds to the join of the corresponding points. Also points which divide these joins proportionally are corresponding points.

In Fig. 441 prove that if G and g divide AC and ac in the same ratio, they are corresponding points, i. e. OgG is straight and OG/Og = k.

3. [SF.3]. In similar figures corresponding lines intersect at corresponding points.

For if Flies on both AC and BE, then by SF.2 the corresponding point flies on both ac and be, and is therefore their intersection. 4. [SF.4]. In similar figures corresponding angles are equal.

To prove AFB

=

Lafb, prove that ▲ AFB | afb. [For a further set of allied theorems see §§ 336-340.]

EXERCISES CXVI.

PRACTICAL GEOMETRY.

1. The sides of a triangle are proportional to 7, 10, and 13. Find the angles.

2. Two angles of a triangle measure 55° and 80° respectively. Express the ratio of the two opposite sides as a decimal fraction. 3. The medians of a triangle are in the ratios 2:3: 4, and the sides bisected by these medians are in the ratios xy: 100. Find x and y.

4. Construct a triangle whose sides measure 2", 2.5", and 3". Construct the locus of a point whose distances from the first two sides are in the ratio 3:5. Measure the two segments into which this locus divides the third side.

5. Construct a triangle whose sides measure 3 cms., 4 cms., 5 cms. Find a point within the triangle whose distances from the sides are proportional to the sides. Measure these distances.

6. Construct a triangle ABC having BC = 3", LB = 50°, ≤ C = 70°. Draw the locus of a point whose distances from the two sides AB, AC (measured parallel to the base) are in the ratio 2:3. At any point of the locus find the ratio of the perp. distances from the two sides.

RIDERS.

7. Any line parallel to the base of a triangle is bisected by the median.

8. The diagonals of a trapezium cut each other in the same ratio.

9. ABC is a triangle, having the angle BAC a right angle, and CD and CE the internal and external bisectors of the angle at C, cutting AB in D and E respectively. Show that AC is a mean proportional between AD and AE.

10. In a right-angled triangle the perpendicular on the hypotenuse is a fourth proportional to the hypotenuse and the other two sides.

11. Prove that the medians of a triangle divide each other into segments that are in the ratio of 1: 2. (Use Chaps. II and XX only.) 12. The straight lines which join corresponding angles of two similar triangles whose corresponding sides are parallel will meet in a point.

13. ABC is a triangle, and a perpendicular is drawn from A to the opposite side, meeting it at D between B and C. Show that, if AD is a mean proportional between BD and DC, BAC is a right angle.

14. If a square DEFG is inscribed in a right-angled triangle ABC so that a side DE of the square lies along the hypotenuse BC, prove that DE is a mean proportional to BD and EC.

15. Two circles with centres A and B touch externally at C. P is a point such that PC bisects APB. If PD, PE are tangents to the circles, prove that PC bisects LDPE.

16. Determine a point whose perpendicular distances from the three sides of a triangle are in the ratio 1: 2: 3.

17. A square is described with one side always along a given line and one corner always on another; find the locus of the vertex which lies on neither.

18. If two triangles ABC and DEF are on equal bases BC and EF, and between the same parallels AD and BF, any straight line drawn parallel to AD will cut off equal areas from ABC and DEF.

19. If two circles touch one another externally their common tangent is a mean proportional to their diameters.

20. If one of the sides of a right-angled triangle is double the other, prove that the perpendicular from the right angle to the hypotenuse divides it in the ratio 4:1.

21. The radius of a circle is a mean proportional between the segments of any tangent made by its point of contact and a pair of parallel tangents.

22. BC is a diameter of a circle and A is any point on the circumference. A point D is taken in BC and a line DE is drawn perpendicular to BC meeting BA, the circle, and CA in E, F, G respectively. Prove that DF2 = DE.DG.

23. Given the base and the vertical angle of a triangle, find the locus of the point of intersection of the medians.

24. Enunciate and prove a converse of Theorem P.6.