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A D BD
... BD X CE
A D X BC.
Again, in the A ABE and B C D,
LABE= 2 DBC,
ZBA E= BDC,
$ 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,
А В A E
BD (A E + C E) = A B X C D + A D X BC,
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.
302. To divide a given straight line into equal parts.
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.
PROPOSITION XXII. PROBLEM.
303. To divide a given straight line into parts proportional to any number of given lines.
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.
Draw FB. From E and C draw E K and CH || to FB.
K and H are the division points required.
А К АН HK KB
CE (a line drawn through two sides of a A ll to the third side divides those
.:. A H : HK : KB : :AC : CE : EF.
Substitute m, n, and o for their equals AC, CE, and EF.
AH : HK : KB : : m :n : 0.
Q. E. F.
304. To find a fourth proportional to three given straight lines.
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.
AC : AB ::CS : BF,
$ 275 (a line drawn through two sides of a A ll to the third side divides those sides
Substitute m, n, and o for their equals A C, A B, and C S.
m :n ::0 : BF.
Q. E. F.
PROPOSITION XXIV. PROBLEM.
305. To find a third proportional to two given straight lines.
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.
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.
(a line drawn through two sides of a A ll to the third side divides those sides
Substitute, in the above equality, A C for its equal B D;
AB: AC :: AC : CE.
Q. E. F.