Art. 110. NOTE.

Euclid's method of stating Proposition 29 amounts to the insertion of additional conditions in the statement here given, the effect of which is to exclude the second alternative in those cases in which the two alternatives are really distinct.

It is as follows:

If two triangles have one angle of the one equal to one angle of the other, and the sides about two other angles proportionals; then if each of the remaining angles be either less or greater than a right angle, or if one of them be a right angle, the triangles are similar and have those angles equal about which the sides are proportionals.

Hence the additional conditions are that ACB and DFE are both less than a right angle, or both greater than a right angle, or one of them is a right angle.

If they are both less than a right angle, their sum is less than two right angles.

If they are both greater than a right angle, their sum is greater than two right angles.

In neither of these cases can the second alternative hold.

Hence the first alternative must hold, and the triangles are similar; the angles between the proportional sides being equal.

If next one of the two angles ACB, DFE is a right angle, then, whichever alternative hold, the other angle is a right angle, hence the remaining angles are equal, and the triangles are similar.

This is the case in which the two alternatives are not really distinct.

Art. 111. EXAMPLE 38.

If B and C are the centres of two circles, and A the point of intersection of their internal or of their external common tangents, and if APQ be any straight line through A cutting the first circle at P and the second at Q, prove that the angles APB, AQC are either equal or supplementary.

Art. 112. PROPOSITION XXX. (Euc. VI. 18.)

ENUNCIATION. On a given straight line to describe a rectilineal figure similar and similarly situated to a given rectilineal figure.

Let А ̧В ̧Ñ‚Ð ̧Е be the given rectilineal figure; it is required to describe a similar figure on the given straight line A,B,, so that A,B1 and A,B, may be corresponding sides of the figures.

1

2

2

A,B, an angle equal to A‚‚С1,

2

A, B, an angle equal to B1A,C1.

2

A,C2 an angle equal to à ̧Ñ1D1,
A,C2 an angle equal to C11D1.

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1

At D, draw a straight line making with D,A, an angle equal to and at A, draw a straight line making with AD, an angle equal to D1A, E. Let these straight lines meet at E2.

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1 1

2

2

It will be proved that АВ1СD ̧Е and А‚„C,D,E, are similar figures, and that A1B1 and A,B, are corresponding sides.

2

In the three pairs of triangles in Figure 89, viz. :—Ã ̧В1C1 and А ̧Ð1⁄2Ñ1⁄2‚ Ã ̧Ñ¡Dı and A‚Ñ1⁄2Ð1⁄2, A¡DĒ ̧ and А„DĒ, let the equal angles be marked with the same

2

2,

numbers. Then in each pair of triangles two angles of the one triangle are respectively equal to two angles in the other triangle. Therefore the remaining angles are equal. Let these be marked with the same number.

H. E.

9

Then it is at once apparent that the angles at A1, B1, C1, D, E, of the figure A,B,C,D,E, are respectively equal to the angles at A2, B2, C2, D2, E2 of the figure A‚B¿Ñ‚D¿E¿; so that the first set of conditions (see Art. 91) for the similarity of

2

1

2

the two figures is satisfied.

Next the triangles A,B,C1 and A,B,C2 are equiangular and therefore by Prop. 26 are similar.

In like manner the triangles A1Ñ‚Ð ̧ and à ̧Ñ‚Ð2 are similar; and the triangles A1D1E, and A,D,E, are similar.

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2

From these pairs of similar triangles follow the relations

B1С1: B2С2 = С1A1: С2Ã1⁄2= Â ̧Â ̧ : А1⁄2Ð1⁄2

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1

(1),

(2), (3).

Hence the second set of conditions (see Art. 91) for the similarity of the two figures is also satisfied.

2 2

2 2

Hence the two figures A,B,C,D1Е1 and AВ2СDE, are similar figures, and A1В1 and AВ are corresponding sides. Therefore the figures are similarly described on A,B, and A,B2 (see Art. 94).

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2 2

Art. 113. PROPOSITION XXXI. (Included in Euc. VI. 20.)

ENUNCIATION. Two similar rectilineal figures may be divided into the same number of triangles such that every triangle in either figure is similar to one triangle in the other figure.

and

Let А ̧ÂÎÑ¡D1‚ ÂÂ1⁄2Ñ‚Ð1⁄2 be two similar figures, such that

2

2

1

Let O1 be any point in the plane of  ̧ ̧С1D1, and join О11, О1B1, О1C1, O1Dı. Through A, draw a straight line making with A,B, an angle equal to  ̧ ̧0.

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2

Through B2 draw a straight line making with B,4, an angle equal to ‚О1. Let these two straight lines meet at 0.

Join O2C2, O2D2.

2

It will be proved that the triangles O1A1B1, О¿Ã‚Â1⁄2 are similar; that the triangles O1B1C1, О2B2C2 are similar, and so on.

2 2

Hence by Prop. 28 the triangles 01B1C1, O2BC, are similar.

2

In like manner the triangles 0 ̧¤ ̧Ð ̧ and О1⁄2Ñ‚D2 can be proved to be similar; and also O, D1A1, ODA, can be proved to be similar.

2 2

So the two similar figures are divided up into the same number of triangles, such that every triangle in either figure is similar to one triangle in the other figure.

The point O1 in the one figure corresponds to the point O, in the other figure. Since O1 is any point in the one figure, it follows that to every point in one of the figures corresponds one and only one point of the other figure.

Art. 114. COROLLARY.

If in Figure 90 the first figure be placed on the second so that O1 falls on 02, О11 falls along О2A2, О1B1 falls along 02B2, it can be shown that the sides of the first figure will then be parallel to and in the same direction as the corresponding sides of the second, and that the distances from 01 or 02 to a point on either figure along any straight line are in the ratio of similitude of the figures.

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1

2

Iƒ 0, be placed on 02, О11 along A2O2 produced through О2, and О ̧Â1 along B2O2 produced through O2, the sides of the first figure will then be parallel but in the opposite direction to the corresponding sides of the second figure.

2

When the two figures have been placed as described in either of the two preceding cases, then the point 02, with which O1 coincides, is called a centre of similitude of the two figures.

The term centre of similitude is not however restricted to rectilineal figures. (See Art. 115, Ex. 40 below.)

Art. 115. EXAMPLES.

39. If two similar rectilineal figures are placed so that two consecutive sides of one figure are respectively parallel and both in the same direction as, or both in the opposite direction to, the corresponding sides of the other figure, then each side of the one figure will be parallel to the corresponding side of the other figure, and the straight lines joining corresponding angular points of the two figures are all parallel or meet in a point; and in the latter case the distances from that point along any straight line to the points where it meets corresponding sides of the figures are in the ratio of similitude of the figures.

What is the ratio of similitude when the lines joining corresponding angular points. are parallel?

40. If the straight line joining the centres A, B of two circles be divided internally and externally in the ratio of the radii of the circles, (the segment of the line AB terminated at A corresponding to the radius of the circle whose centre is A), then show that the points of division may be regarded as centres of similitude of the circles.

Art. 116. PROPOSITION XXXII. (Euc. VI. 8.)

ENUNCIATION. If a right-angled triangle be divided into two parts by a perpendicular drawn from the vertex of the right angle on to the hypotenuse, then the triangles so formed are similar to each other and to the whole triangle; the perpendicular is a mean proportional between the segments of the hypotenuse ; and each side is a mean proportional between the adjacent segment of the hypotenuse and the hypotenuse.