Imágenes de páginas
PDF
EPUB

The figures ABHKL and CDEFG are, therefore, equiangular and they have their fides about the equal angles, alfo, proportional.

For, fince the triangles ALB, CGD are equiangular, AL will be to LB as CG to GD (VI. 5.); or AL to CG as LB to GD (V. 15.)

And, in like manner, LK will be to LB as GF to GD (VI. 5.); ́or LK to GF as LB to GD (V. 15.)

But ratios which are the fame to the fame ratio are the fame to each other (V. 11.); whence AL will be to CG as LK to GF; or AL to LK as CG to GF (V.15.)

And, in the fame manner, it may be 'fhewn, that the fides about the angles K, H, E, A, are proportional to the fides about the angles F, E, D, C.

The figure ABHKL is, therefore, fimilar, and fimilarly fituated with the figure CDEFG (VI. Def. 1.); and it is defcribed upon the right line AB, as was to be done.

H

PROP. XVI. THEOREM,

Equiangular, or fimilar triangles, are sto each other as the fquares of their homologous

fides.

M

Let ABC, DEF be two fimilar triangles, of which the fides AB, DE are homologous; then will the triangle ABC

be to the triangle DEF as the fquare of AB is to the square

[ocr errors]

of DE.

For, on AB, DE defcribe the squares AL, DN (II. 1.), and let fall the perpendiculars CG, FH (I. 12.)

Then, fince the triangles ABC, DEF are fimilar (by Hyp.), AC will be to AB as DF to DE (VI. Def. 1.); or AC to DF as AB to DE (V. 15.)

And, because the triangles AGC, DHF are equiangular, AC will be to CG as DF to FH (VI. 5.); or AC to DF as CG to FH (V. 15.)

But ratios which are the fame to the fame ratio, are the fame to each other (V. 11.); therefore CG is to FH as AB to DE; or CG to AB as FH to DE (V. 15.)

And fince triangles which have the fame base, are to each other as their altitudes (VI. 2.), the triangle ABC is to the triangle AKB as CG is to AK, or AB.

In the fame manner it may be shewn, that the triangle DEF is to the triangle DME as FH is to DM, or DE.

But CG has been fhewn to be to AB as FH is to DE; therefore the triangle ABC is to the triangle AKB as the triangle DEF is to the triangle DME (V. 11.)

And fince the fquare AL is double the triangle AKB (I. 32.), and the square DN is double the triangle DME, the triangle ABC will be to the triangle DEF as the fquare AL is to the fquare DN (V. 13 and 15.)

Q. E. D.

PRO P. XVII. THEOREM.

Similar polygons are to each other as the fquares of their homologous fides.

E

H

Let ABCDE, FGHIK be fimilar polygons, of which AB, G are homologous fides; then will the polygon ABCDE be to the polygon FGHIK as the square of AB is to the fquare of FG.

For join the points BE, BD, GK and GI:

Then, fince the angle A is equal to the angle F, and AB is to AE as FG is to FK (VI. Def. 1.), the triangles EAB, KFG will be equiangular, or fimilar (VI. 6.)

And if, from the equal angles AED, FKI, there be taken the equal angles AEB, FKG, the remaining angles BED, GKI will also be equal to each other.

But ED is to KI as EA is to KF (VI. Def. 1. and V. 15.), and EA is to KF as EB to KG (VI. 5. and V. 15.) 3 whence ED will be to KI as EB is to KG (V. 11.)

Since, therefore, the angles BED, GKI are equal to each other, and the fides about them are proportional, the triangles BED, GKI will, alfo, be equiangular, or fimilar (VI. 6.)

And, in the fame manner, it may be fhewn, that the triangles BCD, GHI are equiangular, or fimilar.

[blocks in formation]

But fimilar triangles are as the fquares of their like fides (VI. 16.); whence the triangle EAB is to the triangle KFG as the fquare of EB is to the fquare of KG.

And, for the fame reason, the triangle EBD is to the triangle KGI as the fquare of EB is to the square of KG.

But ratios which are the fame to the fame ratio, are the fame to each other (V. 11.); whence the triangle EAB is to the triangle KFG as the triangle EBD is to the triangle KGI.

And in the fame manner it may be fhewn that the triangle EBD is to the triangle KGI as the triangle DBC is to the triangle IGH.

The triangle EAB, therefore, is to the triangle KFG, as the triangle EBD is to the triangle KGI, and as the triangle DBC is to the triangle IGH (V. 11.)

And fince the fum of the antecedents is to the sum of the confequents as the firft antecedent is to its confequent (V. 16.), the polygon ABCDE will be to the polygon FGHIK as the triangle EAB is to the triangle KFG.

But the triangle EAB is to the triangle KFG as the fquare of AB is to the fquare of FG (VI. 16.); whence the polygon ABCDE is alfo to the polygon FGHIK as the quare of AB is to the fquare of FG.

Q. E. D.

PRO P. XVIII. THEOREM.

Parallelograms and triangles, having two equal angles, are to each other as the rectangles of the fides which are about those angles.

[blocks in formation]

Let AB, AG be two parallelograms, having the angle DAF equal to the angle GAE; then will AB be to AC as the rectangle of DA, AF is to the rectangle of GA, AE.

For let the fides DA, AE be placed in the fame right line, and complete the parallelogram AK.

Then, because the angles DAF, FAE, are equal to two right angles (I. 13.), and the angle FAE is equal to DAG (I. 15.), the angles DAF, DAG are also equal to two right angles; whence FG is a right line (I. 14.)

And fince parallelograms, of the fame altitude, are to each other as their bafes (VI. 1.), the parallelogram AB is to the parallelogram AK as AD is to AE.

But AD is to AE as the rectangle of AD, AF is to the rectangle of AE, AF (VI. 2. Cor. 2.); therefore AB is to AK as the rectangle of AD, AF is to the rectangle of AÉ, AF (V. II.)

And in the fame manner it may be fhewn, that AC is to AK as the rectangle of AG, AE is to the rectangle of

[blocks in formation]
« AnteriorContinuar »