then As ABC and DEF, are eq. ang.
LABC = DEF, and ▲ C = ▲ F.
If ABC ≤ DEF, let ▲ ABC >
DEF; and make ABG = DEF.
And A = 2 D, ▲ ABG = /DEF,
and AGB = < DFE;
..A ABG is eq. ang. to ▲ DEF;
and .. AB: BG = DE: EF;

but as DE EF so AB: BC,
.. AB : BC = AB: BG;
and AB: BC = AB : BG,
.. BC = BG;

3

and..

7 H. 13. I.

BCG.

CASE II. Let s C&F be each a rt. ▲ ;

4 Conc.

=

BG,

G

:: ≤ C =
ZBGC;

rt. Li

but C

BGC rt. ;

.. two s of ▲ BGC, are ← 2rt.

is impossible;

and .. ▲ ABC is eq. ang. to ▲ DEF, as

in Case I.

CASE III.

Conc.

Let one of thes, C, F, namely C, be a rt. Li

SCH. When two angles are both greater, or both less than right angles they are both said to be of the same affection; and in enunciating this Prop, instead of "both greater or both less than right angles," it is not unusual to say, "both of the same affection."

USE AND APP.-In Book I. Propositions 4, 8 and 26 contain the criteria of the equality of two triangles; and in Book VI. Propositions 4, 5, 6 and 7 may be classed together as giving the conditions on which we declare the similarity of two triangles. Equality in triangles is absolute, not in area only ;— but the similarity is likeness of shape, not identity of size.

The criteria of similarity are;

1o,—The equality of the three angles, 4, VI.;

2°. The identity of the ratios of the respective sides, 5, VI.;

3°.-The equality of two angles, one in each triangle,-and the identity of the ratios of the containing sides, 6, VI.;

4°. The identity of the ratios of two sides in each triangle,-the equality of an angle in each opposite one pair of homologous sides;-and each of the remaining angles opposite the other pair of homologous sides less than a right angle, or one of them a right angle.

Universally, if triangles fulfil any one of these four conditions of similarity, the sides about the equal angles are proportional.

PROP. 8.-THEOR. (Important.)

In a rt. angled triangle, if a perpendicular be drawn from the right angle to the base; the triangles on each side of it are similar to the whole triangle and to one another.

DEM. AX. 11, I. All rt. s are equal to one another. 32, I. 4, VI. Def. 1, VI.

:: BAC = LADB, & LB com.;

.. rem, ACB = rem. BAD

A

.. AABC is eq. ang. to ▲ ABD, and the sides propls.;

.. AABC is similar to ▲ ABD.

So, ▲ ADC is eq. ang. and sim. to ▲ ABC; and .. ▲ ABD is eq. ang. & sim. to ▲ ACD, which is eq. ang. and sim. to ▲ ABC. Therefore, in a rt. angled triangle, &c.

Q. E. D.

COR. 1. The perpendicular, A D, from the vertex of the rt. L BAC to the opposite side BC, is a mean proportional between the segments BD, DC of this side; and also each of the sides, BA, AC, including the rt. is a mean proportional between the opposite side BC, and the segment of it, BD, or DC adjacent to that side, BA or AC.

D. 1 H.

Def. 1, VI.
Obs. Def. 10, V.

2 H. 4, VI.

3 H. 4, VI.

:: ▲ ADB is sim. to ▲ ADC,

.. BD : DA = DA: DC;

i. e. DA is a mean proportional between
BD and DC.

Also, ▲ ABC is sim. to ▲ DBA,
.. BC: BA = BA: BD;

i. e. A B is a mean proportional between
BC and BD.

And▲ ABC is sim. to ▲ ACD,

.. BC: CA = CA: CD;

i. e. AC is a mean proportional between BC and CD.

COR. 2. The segments BD, CD, of the hypotenuse, made by the perp. AD, are to one another as the squares on the sides of the BA2: AC2.

rt. Li

by D. 2 & 3 of Cor. 1. 8, VI. BD. BC: CD. BC = BA2
: AC2;

if we divide the 1st & 2nd terms of the analogy by BC,
then BD CD = BA2: CA2

COR. 3. The squares on the sides about the rt.

and on the

hypotenuse are to each other as the segments of the hypotenuse, made by the perp. AD, and the hypotenuse itself.

·.· BD.BC: CD.BC : BC.BC = AB2 : CA2 : B C2;
.. on dividing the three terms by BC,
BD: CD: BC = AB2: CA2: BC2.

N. B.-Several other subsidiary Corollaries might be added,—but the most important deductions have been given, and we subjoin only ;

COR. 4. If the base of a triangle, BC, the two sides, AB, AC, and the perpendicular AD, be four proportionals, the triangle must be right angled.

. in ▲ ABC, and in one of the component
AS ABD, two sides are proportionals;
and the s opp. one pair of homologous sides
are equal;

and also of the s opp. the other pair of
homol. sides one is a rt. ;

.. the whole ▲ ABC is sim. to the component ▲ ABD;

and .. also ▲ ABC is rt. angled.

SCH. 1. The 8th Proposition, and the deductions that may be made from it are particular cases of a more general principle; namely,

If from the vertex, B, of a ▲ ABC, two lines BD, BE. be drawn to the base, AC, making the angles at the base, BDA. and BEC, or their supplements each equal to the vertical angle ABC; then the As BDA, BEC, formed by those lines, and by the segment DA or EC which each cuts off, shall be similar to the whole triangle and to one another.

E. 1 Hyp.

2 Conc.

D. 1 Remk.

Let ABC be a A, and from vertex B, let there be drawn BD, BE, making ▲ BDA = / BEC = / ABC; then ▲ BED is isosc.; and ABDA is eq. ang. to ABEC which is eq. ang. to ▲ ABC.

When ABC > rt., s BDA and BEC are ext. s. at the base of the isosc. ▲ BDE;