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3. Prove that AD X DC + Fco + FES = DF + FE',
and that these, 4.
= EB + BDS, 5. that AD X DC + EB? – EB + BD, 6. that AD X DC = BD?.
Cor. If from a point without a circle two right lines, as AB, AC, be drawn cutting the circle, then AB X
AE = AC X AF.
The proof of this is merely that AB X AE and ac X AF are each = ADP, and :. = each other.
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 2, 2. that DE' = AD X DC, and :: = DB”, 3
that .:. DE = DB, 4. that (in AS DFB, DFE) Z DEF = _ DBF, 5. that DBF is a right Z, 6. that de touches the o.
Proved by showing, from the construction, that AC = CE; and CE = D.
Steps of the Demonstration. 1. Prove that _ ABC = _ HAC, and :: = DEF, 2. that similarly ACB = _ DFE, 3. that A ABC is equiangular to the A DEF.
1. Prove that all the Zs at A, B, and c are rt. ZS, 2. that ZS AMB + AKB = 2 rt. Zs, and .. =
ZS DEG + DEF, 3. that / LMN = L DEF,
that similarly _ LNM = L.DFE, 5 that A mln is equiangular to A DEF.
Problem. To inscribe a circle in a given triangle.
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.
Problem. To describe a circle about a given triangle.
(In the construction of the figure, prove that the perpendiculars DF and EF will meet, as in F).
1. Prove that (in As ADF, BDF) base BF = base AF, 2. 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.
Problem. To inscribe a square Bk in a given circle.
Steps of the Demonstration. 1. Prove that (in AS ABE, AED) base AB = base
AD, 2. 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, 5. that :: ABCD is a square, and is inscribed
Steps of the Demonstration. 1. Prove that s at A are right Zs; and that simi
larly Zs at B, C, and D are right Zs.