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345. If any chord is drawn through a fix within a circle, the product of its segments stant in whatever direction the chord is dra B

Let any two chords AB and CD intersect a

OAX OBOD × OC.

Draw AC and BD.

In the AAOC and BOD,

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(two are similar when two 4 of the one are equal to two d OA, the longest side of the one, :OD, the longest side of the other, :: OC, the shortest side of the one, : OB, the shortest side of the other. .. OAX OB = OD × OC.

346. SCHOLIUM. This proportion may be written

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that is, the ratio of two corresponding segments is the reciprocal of the ratio of the other two corr segments In this case the segments are said to be re proportional.

347. If from a fixed point without a circle a secant is drawn, the product of the secant and its external segment is constant in whatever direction the secant is drawn. 072

and

H

B

Let OA and OB be two secants drawn from point 0.

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(two are similar when two of the one are equal to two of the other).

Whence

OA, the longest side of the one,

: OB, the longest side of the other,
::OD, the shortest side of the one,

: OC, the shortest side of the other,
.. OA × OC=0B × OD.

§ 295

Q. E. D,

REMARK. The above proportion continues true if the secant OB turns about O until B and D approach each other indefinitely. Therefore, by the theory of limits, it is true when B and D coincide at H. Whence, QA × OC= OH2.

This truth is demonstrated directly in the vort theorem

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348. If from a point without a circle a sec a tangent are drawn, the tangent is a mean tional between the whole secant and the segment.

M

B

Let OB be a tangent and OC a secant dra the point 0 to the circle MBC.

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.. A OBC and OBM are similar, (having two of the one equal to two of the other

Whence

OC, the longest side of the one,
: OB, the longest side of the other,
:: OB, the shortest side of the one,
: OM, the shortest side of the other.

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349. The square of the bisector of an angle of a triangle is equal to the product of the sides of this angle diminished by the product of the segments determined by the bisector upon the third side of the triangle.

B

D

E

Let AD bisect the angle BAC of the triangle ABC.

To prove
Proof.

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Circumscribe the O ABC about the ▲ ABC.

Produce AD to meet the circumference in E, and draw EC.
Then in the A ABD and AEC,

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(two ▲ are similar if two ▲ of the one are equal respectively to two of the other).

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(the product of the segments of a chord drawn through a fixed point in

Whence

a O is constant).

·. AB× AC= AD2 + DB× DC.

AD

ABX AC — DB × DC.

Q. E. D.

mi

350. In any triangle the product of two equal to the product of the diameter of the scribed circle by the altitude upon the third

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Let ABC be a triangle, AD the altitude, the circle circumscribed about the triangle Draw the diameter AE, and draw EC.

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..A ABD and AEC are similar,

(two rt. A having an acute of the one equal to an acute 4

Whence

are similar).

AB, the longest side of the one,
AE, the longest side of the other,
:: AD, the shortest side of the one,
: AC, the shortest side of the other.
.. ABX AC= AE× AD.

NOTE. This theorem enables us to compute the length of a circle circumscribed about a triangle, if the lengths of th of the triangle are known.

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