PROP. LXX. THEOR. 15. 5 Eu. Equimultiples of two quantities have the same ratio as the quantities themselves.

Let ma and mb be equimultiples of a and b; then will a:b::ma: mb.

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Triangles and parallelograms of the same altitude are to each other as their bases.

In str. line BD take any number of equal parts or units BG, GC, CH, &c. From pt. À without the line BD draw AB, AG, AC, AH, &c. : also, draw AF, EC, FD respectively || BD, 'AB, AC; and the AS ABC, ACD having the same altitude, viz. the perpendicular drawn from A to BD; then will base BC : base CD :: A ABC : AACD :: DM ABCE: OMACDF.


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each other;

_ No. of As in No. of parts in BC =

- ABC, And

D No. of As in
No. of parts in CD = ACD,
No. of parts in BC No.of Asin ABC
No. of parts in CD No. of As in ACD


i.e. CD=2ACD or, Base BC: Base CD :: A ABC: AACD

JOMABCE: 0 Prop. 70.

| ACDF. Therefore triangles, &c.

PROP. LXXII. THEOR. If a straight line be drawn parallel to one of the sides of a triangle, it shall cut the other sides proportionally: and if the sides be cut proportionally, the straight line which joins the points of section shall be parallel to the remaining side of the triangle.

1st. Draw DE || BC one of the sides of the AABC; then shall DB: AD:: EC:AE; also AB: AC :: AD: AE.

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

.. DE || BC. Wherefore if a str. line, &c. .


PROP. LXXIII. THEOR. Equiangular triangles have their correspond

ing sides about the equal angles in the same proportion.

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

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Conceive ADEF so applied to AABC, that
D may coincide with A, and DE the lesser fall
upon AB the greater.

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Prop. 4

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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 LA=_Dand AC: AB :: DF: DE; then will the As be equiangular, and the Ls A, B, C= _s D, E, F; ea. to ea.

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