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For the triangles HSP, DPS being each of them the given rectangle ABDC, therefore by taking away DBRVC common, there will remain the triangle RAV

SBR + VCP.
The construction is the same for any parallelogram.

PROP. 102. PROB.

To find a point Pin an indefinite right line PQR

touching a given circle in Q, al which erecting PD at right angles to PR cutting the circle in E and D, bisecting ED in I, and drawing Il parallel to PR meeting the circle in F, so that drawing DF PF, the triangle DPF 80 formed may be a maximum.

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Fig. 45.

Draw a tangent RCA at right angles to RP, then from R through the centre of the given circle draw RD, let fall DP at right angles to RQ then P is the point required.

For, draw a tangent ADB and complete the [? CP which is the greatest rectangle that can be inscribed in the right angled triangle ARB (prop. on b. 1.) and .. much greater than any other rectangle, having its angular point in the periphery of the given circle, because any rectangle so formed, is less than its similar one inscribed in the triangle, whence it is evident that the triangle PFD being half the square PC is the greatest possible.

PROB. 103. PROB.

To determine the locus of the point P so that if from

a given point E, EP be drawn and EL perpendicular to it to meet the right line 1 R given in position the angle ELP may be always equal to a given acute

angle. Fig. 98.

Draw EQ at right angles and ET parallel to MR and make QEN = to the given angle (N being in MR) through N draw a right line parallel to EQ, and it shah be the required locus.

In NT assume any point P, and draw PL according to the proposition; on PL as diameter, describe a circle which must pass through E and N, because the angles at those points are right, and the angle QEN = ENP; but the angles ENP, ELP are = (prop. 23. b. 3. Elr.) ::. ENP = ELP = QEN the given angle, (by construction) and the point P is always in the same right line TN given by position,

Besna

THE

ELEMENTS

OF

PLANE GEOMETRY,

COMPRISING

THE DEFINITIONS OF THE FIFTH BOOK, AND THE SIXTH BOOK IN GENERAL TERMS.

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PRINTED BY R. GRAISBERRY,
FOR HODGES AND MʻARTHUR, 21, COLLEGE-GREEN.

1824,

1931.E.IS 2)

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TO STUDENTS.

OF the former parts of this work that have been already before the public, a new and much improved edition shall be published in the course of the ensuing year.

There is also in a forward state a collection of Algebraic questions; comprising examples on each rule, in order, but most particularly on the application of Algebra to Geometry, which is certainly a desideratum at the

present time.

The Author having completely made himself up in the undergraduate course, intends to devote part of his time, in future, to pupils. His terms shall be found moderate; two or three reading in the same Class will be instructed at proportionately reduced prices. Application to be made to him personally, or by letter directed to him at 16, College.

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