Prop. 34.
Cor.

Prop. 71.

Join BE, CD:

As BDE, CDE, are on the same base DE, and between the same ||s DE and BC:

.. ABDE=ACDE,

ABDE ACDE

but's having the same alt., are as their

Ax. 1.

Prop. 67.
Prop. 68.

i. e.

DB: AD:: EC: AE.

Also DB: EC:: AD: AE,

.. DB+AD: EC+AE:: AD: AE,
AB: AC:: AD: AE.

i. e.

2nd. Let the sides AB, AC of the

ABC, be cut proportionally in pts. D, E, so that DB: AD :: EC: AE, or AB:AC::AD:AE; join DE: then shall DE || BC.

But As BDE, CDE are on the same base
DE, and ... are between the same ||s;

DE || BC.

Wherefore if a str. line, &c.

PROP. LXXIII.

THEOR.

Equiangular triangles have their corresponding sides about the equal angles in the same proportion.

Let ABC, DEF be equiangular As, having Zs A, B, C, respectively equal to s D, E, F, ea. to ea.; then shall AB: AC:: DE: DF; or AB: DE:: AC: DF.

Conceive DEF so applied to

E

ABC, that

D may coincide with A, and DE the lesser fall upon AB the greater.

Prop. 36.

Prop. 4

Def. 3. pa. 89.

Similarly

AB: BC:: DE: EF.

Therefore equiangular triangles, &c.
COR.-Equiangular triangles are similar.

PROP. LXXIV. THEOR. 6. 6 Eu.

Two triangles are equiangular, when an angle in the one is equal to an angle in the other, and the sides about the equal angles are proportionals.

In As ABC, DEF, let A= /D and AC: AB:: DF: DE; then will the As be equiangular, and the s A, B, C = ≤s D, E, F ;

../ B=/ E, ≤C=/F,

i. e. s A, B, C= s D, E, F; ea. to ea.
..As ABC, DEF are equiangular.

Wherefore if two triangles, &c.

PROP. LXXV. THEOR.

8. 6 Eu.

In a right angled triangle, if a perpendicular be drawn from the right angle to the opposite side, the triangles on each side of the perpendicular are similar to the whole triangle, and to each other.

Let ABC be a rt. angled, having the rt. ▲ BAC; draw AD BC; then the ◇s ABD, ADC are similar to the whole ABC, and to one another.

In the As ABC, ABD,

BAC ADB, ea. being a rt. L,

B common to both

s;

G

Prop. 4.

Hyp.

Prop. 72.

Prop. 28.

... rem. ACB =rem. BAD;

And As ABC, ABD are equiangular; also the corresponding sides are proportionals (Prop. 73); and the 's are similar (Cor. Prop. 73).

In the same way it may be shown that ADC is equiangular and similar to ABC and also to ABD.

Therefore in a rt. angled A, &c.

COR. The perpendicular AD is a mean
proportional between the segments BD, DC,
of the base BC. And each of the sides AB,
AC, is a mean proportional between the base
BC and the segment adjacent to the side.
For in the

As, BDA, ADC, BD: DA :: DA: DC;
As, ABC, DBA, BC : BA :: BA: BD;
As ABC, ACD, BC : CA :: CA : CD.

PROP. LXXVI. THEOR. 33. 6 Eu. In the same or equal circles, angles at the centres have the same ratio which the arcs on which they stand have to each other.

In the ABD, whose centre is L, take any No. of - /s DLK, KLI, ILH, &c. ; theu shall arc DC: arc CB:: DLC: CLB.