BOOK VI.

INTRODUCTORY REMARKS.

THE chief subject of this Book is the Similarity of Rectilinear Figures.

DEF. I. Two rectilinear figures are called similar, when they satisfy two conditions :

I. For every angle in one of the figures there must be a corresponding equal angle in the other.

II. The sides containing any one of the angles in one of the figures must be in the same ratio as the sides containing the corresponding angle in the other figure: the antecedents of the ratios being sides which are adjacent to equal angles in each figure.

Thus ABC and DEF are similar triangles, if the angles at A, B, C be equal to the angles at D, E, F, respectively, and if BA be to AC as ED is to DF,

and AC be to CB as DF is to FE,
and CB be to BA as FE is to ED.
A

The sides adjacent to equal angles in the triangles are thus homologous, that is, BA, AC, CB are respectively homologous to ED, DF, FE.

It will be shown in Prop. IV. that in the case of triangles the second of the above conditions follows from the first.

In the case of quadrilaterals and polygons both conditions are necessary: thus any two rectangles have each angle of the one equal to each angle of the other, but they are not necessarily similar figures.

N.B.-The very important Prop. xxv. (Eucl. vi. 33) is independent of all the other Propositions in this Book, and might be placed with advantage at the very commencement of the Book.

PROPOSITION I. THEOREM.

Triangles of the same altitude are to one another as their

Let the As ABC, ADC have the same altitude, that is, the perpendicular drawn from A to BD.

Then must ▲ ABC be to ▲ ADC as base BC is to base DC.

In DB produced take any number of straight lines

BG, GH each=BC.

I. 3.

In BD produced take any number of straight lines DK, KL, LM each=DC.

I. 3.

Join AG, AH; AK, AL, AM.

Then CB, BG, GH are all equal,

.. ▲s ABC, AGB, AHG are all equal.

I. 38.

..▲ AHC is the same multiple of ▲ ABC that HC is of BC.

So also,

▲ AMC is the same multiple of ▲ ADC that MC is of DC. And ▲ AHC is equal to, greater than, or less than ▲ AMC, according as base HC is equal to, greater than, or less than base MC.

Now A AHC and base HC are equimultiples

of A ABC and base BC,

and ▲ AMC and base MC are equimultiples

of A ADC and base DC.

I. 38.

.. A ABC is to ▲ ADC as base BC is to base DC. V. Def. 5.

Q. E. D.

COR. I. Parallelograms of the same altitude are to one another as their bases.

Let ACBE, ACDF be parallelograms having the same altitude, that is, the perpendicular drawn from A to BD.

Then must ACBE be to ACDF as BC is to DC.

COR. II. Triangles and Parallelograms, that have EQUAL altitudes, are to one another as their bases.

Let the figures be placed, so as to have their bases in the same straight line; and having drawn perpendiculars from the vertices of the triangles to the bases, the straight line, which joins the vertices, is parallel to that, in which their bases are, because the perpendiculars are both equal and parallel to one another. I. 33.

Then, if the same construction be made as in the Proposition, the demonstration will be the same.

Ex. 1. ABC, DEF are two parallel straight lines; show that the triangle ADE is to the triangle FBC as DE is to BC.

Ex. 2. If, from any point in a diagonal of a parallelogram, straight lines be drawn to the extremities of the other diagonal, the four triangles, into which the parallelogram is then divided, must be equal, two and two.

PROPOSITION II. THEOREM.

If a straight line be drawn parallel to one of the sides of a triangle, il must cut the other sides, or those sides produced, proportionally.

Let DE be drawn || to BC, a side of the ▲ ABC.
Then must BD be to DA as CE to EA.

Join BE, CD.

Then ABDE= ▲ CDE, on the same base DE and between the same ||s, DE, BC.

.. A BDE is to ▲ ADE as A CDE is to ▲ ADE

But A BDE is to A ADE as BD and A CDE is to A ADE as CE ᎠᎪ as CE

BD is to

is to

I. 37.

V. 6.

DA,

VI.1.

is to EA;

VI. 1.

is to EA.

V. 5.

Ex. 1. If any two straight lines be cut by three parallel lines, they are cut proportionally. (N.B.-This is of great use.)

Ex. 2. If two sides of a quadrilateral be parallel to each other, a straight line, drawn parallel to either of them, shall cut the other sides, or these produced, proportionally.

And Conversely,

If the sides, or the sides produced, be cut proportionally, the straight line which joins the points of section must be parallel to the remaining side of the triangle.

Let the sides AB, AC of the ▲ ABC, or these produced, be cut proportionally in D and E, so that

and join DE.

BD is to DA as CE is to EA,

Then must DE be parallel to BC.

The same construction being made,

BD is to DA as CE is to EA,

and BD is to DA as ▲ BDE is to A ADE,

VI. 1.

and CE is to EA as ▲ CDE is to ▲ ADE,

VI. 1.

.. ▲ BDE is to ▲ ADE as ▲ CDE is to ▲ ADE, V. 5.

Ex. 3. If there be four parallel straight lines, two of these lines intercept upon two given lines, of unlimited length, OA, OB, parts proportional to the parts intercepted upon OA, OB, by the remaining two parallel straight lines.

Ex. 4. If the four sides of a quadrilateral figure be bisected, the lines joining the points of bisection will form a parallelogram.

Ex. 5. A quadrilateral figure has two parallel sides: shew that the straight line, joining the point of intersection of its other two sides produced and the point of intersection of its diagonals, bisects the two parallel sides.