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or,

§ 278

A D BD

CE
(the homologous sides of similar S are proportional).

BO'

... BD X CE

A D X BC.

Again, in the A ABE and B C D,

LABE= 2 DBC,

Cons.

§ 203

and

ZBA E= BDC,
(each being measured by t of the arc BC).
..A A B E and B C D are similar,

$ 280 (two A are similar when two ts of the one are equal respectively to two 5

of the other).

Whence A B, the longest side of the one,

: BD, the longest side of the other,
: : A E, the shortest side of the one,
: CD, the shortest side of the other.

or,

§ 278

А В A E

BD CD
(the homologous sides of similar O are proportional).

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BD (A E + C E) = A B X C D + A D X BC,

or

BD X AC=AB XCD + A D X BC.

Q. E. D.

Ex. If two circles are tangent internally, show that chords of the greater, drawn from the point of tangency, are divided proportionally by the circumference of the less.

[blocks in formation]

302. To divide a given straight line into equal parts.

А

B

0

Let A B be the given straight line.

It is required to divide A B into equal parts.

From A draw the indefinite line A 0.

Take any convenient length, and apply it to A 0 as many times as the line A B is to be divided into parts.

From the last point thus found on A 0, as C, draw C B.

Through the several points of division on A O draw lines ll to C B.

These lines divide A B into equal parts,

§ 274 (if a series of lls intersecting any two straight lines, intercept equal parts

on one of these lines, they intercept equal parts on the other also).

Q. E. F.

Ex. To draw a common tangent to two given circles.
I. When the common tangent is exterior.
II. When the common tangent is interior.

PROPOSITION XXII. PROBLEM.

303. To divide a given straight line into parts proportional to any number of given lines.

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Let A B, m, n, and o be given straight lines.

It is required to divide A B into parts proportional to the given lines m, n, and o.

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Draw FB. From E and C draw E K and CH || to FB.

K and H are the division points required.

(AE)

$ 275

А К АН HK KB
For
A E AC

CE (a line drawn through two sides of a A ll to the third side divides those

sides proportionally).

.:. A H : HK : KB : :AC : CE : EF.

Substitute m, n, and o for their equals AC, CE, and EF.

Then

AH : HK : KB : : m :n : 0.

Q. E. F.

[blocks in formation]

304. To find a fourth proportional to three given straight lines.

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Let the three given lines be m, n, and o.

It is required to find a fourth proportional to m, n, and o.

Take A B equal to n.

Draw the indefinite line A R, making any convenient 2 with A B.

On A R take A C=m, and CS=0.

Draw C B.

From S draw SF || to C B, to meet A B produced at F.

B F is the fourth proportional required.

For,

AC : AB ::CS : BF,

$ 275 (a line drawn through two sides of a A ll to the third side divides those sides

proportionally).

Substitute m, n, and o for their equals A C, A B, and C S.

Then

m :n ::0 : BF.

Q. E. F.

PROPOSITION XXIV. PROBLEM.

305. To find a third proportional to two given straight lines.

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Let A B and AC be the two given straight lines.

It is required to find a third proportional to A B and A C.

Place A B and A C so as to contain any convenient Z.

Produce A B to D, making BD= AC.

Join BC.

Through D draw D E ll to B C to meet A C produced at E.

C E is a third proportional to A B and A C.

§ 251

А С

For,

AB
BD

$ 275

CE'

(a line drawn through two sides of a A ll to the third side divides those sides

proportionally).

Substitute, in the above equality, A C for its equal B D;

Then

АВ
AC

AC
CE'

or,

AB: AC :: AC : CE.

Q. E. F.

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