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Theorem. If from a point without a circle there be drawn two straight lines, one of which cuts the circle, and the other meets it; if the rectangle contained by the whole line which cuts the circle, and the part of it without the circle, be equal to the square of the line which meets it, the line which meets shall touch the circle.

Steps of the Demonstration. 1. Prove that fed is a right Z,

that DE’ – AD X DC, and :: = DB',
that .. DE = DB,
that in As DFB, DFE) Z DEF = _ DBF,
that DBF is a right Z,
that de touches the O.

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FOURTH BOOK.

PROPOSITION I.

Problem. In a given circle to place a straight line, not greater than the diameter of the circle.

DProved by showing, from the construction, that AC = Ce; and CE = D.

PROPOSITION II.

Problem. In a given circle to inscribe a triangle equiangular to a given triangle.

si aina

Steps of the Demonstration. 1. Prove that ABC = _ HAC, and .. = _ DEF,

that similarly _ ACB = DFE,
that A ABC is equiangular to the A DEF.

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Steps of the Demonstration. 1. Prove that in As DBE, DBF) side DE = side DF,

by xxvi. I,

2. Prove that similarly DG = DF; and :: DE, DF,

DG = each other, 3.

that a o described with centre d, and dis

tance either of these lines, will touch the sides AB, BC, CA, and :: that O EFG is inscribed in A ABC.

Proposition V.

To describe a circle about a given

Problem. triangle.

Steps of the Demonstration. (In the construction of the figure, prove that the perpendiculars DF and EF will meet, as in F).

ai

1. Prove that (in As ADF, BDF) base BF = base AF,

that similarly cF = AF; and :. AF, BF,

CF = each other, 3. that a o described with centre F, and dis

tance either of these lines, will pass through the extremities of the other two, and be described about the A ABC.

PROPOSITION VI.

Problem. To inscribe a square bk..... in a given circle.

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Steps of the Demonstration. 1. Prove that (in As ABE, AED) base AB = base ad,

that similarly BC, CD = BA, or AD; and ::

AB, BC, CD, DA = each other, that _ Bad is a right Z, that the figure ABCD is rectangular, that :: ABCD is a square, and is inscribed

in o.

PROPOSITION VII.

Problem. To describe a square about a given circle.

Steps of the Demonstration. 1. Prove that <s at A are right Zs; and that simi

larly s at B, C, and d are right Zs.

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