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(18.). If from a given point two straight lines be drawn, containing a given angle, and such that their rectangle may be equal to a given rectilineal

figure, and one of them be terminated by a straight line given in position ; to determine the locus of the extremity of the other.

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Let A be the given point, and BC the line given in position. From A draw AD perpendicular to BC; and draw AE, making with it the angle DAE equal to the given angle; and make AE of such a magnitude that the rectangle AD, AE may be equal to the given figure. On AE as diameter describe a circle AFE; it will be the locus required.

Draw any other line AB, and AF making with it the angle FAB equal to the given angle; join FE. Then the triangles ABD, AFE, being equiangular,

AB : AD :: AE : AF, whence the rectangle AB, AF is equal to the rectangle AD, AE, i, e. to the given figure; and the same may be proved; of any other two lines, similarly drawn from A.

(19.) If from the vertical angle of a triangle two lines be drawn to the base making equal angles with the adjacent sides ; the squares of those sides will be proportional to the rectangles. contained by the adjacent segments of the base.

Let AD, AE be drawn from the vertical angle A making equal angles BAD, EAC with the adjacent sides; then will AB : AC2 :: BDX BE : CDX CE.

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About the triangle ADE describe a circle, cutting AB, AC (produced if necessary) in G and F. Join FG. Then (Eucl. iii. 26.) the arcs GD, FE are equal, .'. (ii. 1. Cor.) FG is parallel to BC;

.. AB : AC :: BG: CF,

and AB2: AC2 :: AB× BG : AC× CF

:: BD BE CDx CE (Eucl. iii. 36.).

(20.) If a line placed in one circle be made the diameter of a second, the circumference of the latter passing through the centre of the former; and any chord in the former circle be drawn through this diameter perpendicularly; the rectangle contained by the segments made by the circumference of the latter circle will be equal to that contained by the whole diameter and a mean proportional between its segments.

Let a line AC, placed in the circle ADC, be the diameter of the circle ABC, whose circumference passes through the centre of ADC. Through any point B let a line DBE be drawn perpendicular to AC; the rectangle DB, BE is equal to the rectangle AC, BF

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Draw CBG. And since the circumference ABC passes through the centre of AGD, .. (ii. 60.) AB is equal to BG, and the rectangle AB, BC is equal to the rectangle GB, BC, i. e. to the rectangle DB, BE. Also the rectangle AB, BC is equal to the rectangle AC, BF, (Eucl. vi. C.), .. the rectangle DB, BE is equal to the rectangle AC, BF.

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(21.) If semicircles-be described on the segments of the base made by a perpendicular drawn from the right angle of a triangle ; they will cut off from the sides, segments which will be in the triplicate ratio of the sides.

From the right angle B let BD be drawn perpendicular to AC; and on AD, DC let semicircles be described, cutting AB, CB in E and F; AE : CF in the triplicate ratio of AB : CB.

Join DE, DF; they are perpendicular to AB, BC respectively; ::. (Eucl. vi. 8. Cor.)

AC : AB :: AB : AD

AB : AD :: AD : AE hence AC : AE in the triplicate ratio of AC : AB. In the same manner it may be shewn that

AC : CF in the triplicate ratio of AC : CB, :. inv. and ex æquo,

AE : CF in the triplicate ratio of AB : CB.

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(22.) If from any point in the diameter of a semicircle a perpendicular be drawn, and from the extremities of the diameter lines be drawn to any point in the circumference, and meeting the perpendicular ; the rectangle contained by the segments which they cut off from the perpendicular, will be equal to the rectangle contained by the segments of the diameter.

From any point D in the diameter AC of the semicircle ABC, let a perpendicular DF be drawn ; and to any point B in the circumference let the lines AB, CB be drawn, meeting the perpendicular

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in E and F; the rectangle FD, DE is equal to the rectangle AD, DC.

Since the angle ABC in a semicircle is a right angle, FBE is also a right angle, and .. equal to FDC; and the vertically opposite angles at E are equal, .-. the angle BFE is equal to ECD, and the triangles FDA, DEC are equiangular,

and FD : DA :: DC : DE, ::. the rectangle FD, DE is equal to the rectangle AD, DC.

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(23.) If from the point of bisection and

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other point in a given arc of a circle, two parallel lines be drawn, the former terminated by the circumference, the latter by the chord of the arc; the rectangle contained by these two lines will be equal to that contained by the lines which join the latter point with each extremity of the given arc.

From C the middle point of the arc ACB, and D any other point, let any two parallel lines CE, DF be drawn, of which CE is terminated by the circumference of the circle, and DF by the chord AB. Join AD, DB; the rectangle CE, DF is equal to the rectangle AD, DB.

Draw DG perpendicular to AB; draw the diameter CH; and join EH. The angle FDG being equal to ECH, and the angles at G and E right angles, the triangles FDG, ECH are equiangular,

.. DG : DF :: EC : CH, whence the rectangle EC, DF is equal to the rectangle DG, CH, i. e. (Eucl. vi. C.) to the rectangle AD, DB. (24.) If two circles cut each other, and from either point of intersection lines be drawn meeting both circumferences ; the rectangles contained by the segments of these lines are to one another in the ratio of the perpendiculars drawn from their intersection with inner cir. cumferences upon the line joining the intersections of the circles.

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Let the two circles ABC, ABE cut each other in A and B; join AB; and from B draw any two lines BC, BD cutting the circles in E, F, C, D; let fall the perpendiculars EG, FH; the rectangles BE, EC and BF, FD are to one another as EG to FH. Join AD, AC. Since the angle AFB= AEB, ..

: AFD= AEC; but ADF= ACE in the same segment, .. the triangles AFD, AEC are similar ;

and EC : EA :: FD: FA.
But EC: EA :: ECX EB : EAX EB

= EG x diameter of the circle ABE,
and FD : FA :: BFX FD : BFX FA

FH x diameter of the circle ABE, whence ECX EB : BFX FD :: EGxD : FHx D ::

[EG : FH.

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(25.) If on opposite sides of any point in the chord of a circle two lines be taken, one terminating in the chord, the other in the chord produced, whose rectangle is equal to that contained by the segments of the chord; and the extremities of the lines so taken be joined to those

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