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PROP. XII. THEOR.
IN obtufe angled triangles, if a perpendicular be drawn
Let ABC be an obtufe angled triangle, having the obtufe
Because the ftraight line BD is divided into two parts in the
to the fquares of BC, CD, and
qual to the fquares of CD, DA: Therefore the fquare of BA
See N. IN every triangle, the fquare of the fide fubtending any of the acute angles, is lefs than the fquares of the fides containing that angle, by twice the rectangle contained by either of these fides, and the ftraight line intercepted between the perpendicular let fall upon it from the oppofite angle, and the acute angle.
a 12. I.
C 47. I.
d 16. r.
Let ABC be any triangle, and the angle at B one of its acute angles, and upon BC, one of the fides containing it, let fall the perpendicular AD from the oppofite angle: The fquare of AC, oppofite to the angle B, is lefs than the fquares of CB, BA by twice the rectangle CB, BD.
First, Let AD fall within the triangle ABC; and because the ftraight line CB is divided into two parts in the point D, the fquares of CB, BD are equal to twice the rectangle contained by CB, BD, and the square of DC: To each of thefe equals add the fquare of AD; therefore the fquares of CB, BD, DA are equal to twice the rectangle CB, BD, and the fquares of AD, DC: But the fquare of AB is equal
to the fquares of BD, DA, because the angle BDA is a right angle; and the fquare of AC is equal to the fquares of AD, DC: Therefore the fquares of CB, BA, are equal to the square of AC, and twice the rectangle CB, BD; that is, the fquare of AC alone is less than the fquares of CB, BA by twice the rectangle CB, BD.
Secondly, Let AD fall with-
fquares of AB, BC are equal to the fquare of AC, and twice the fquare of BC, and twice the rectangle BC, CD: But becaufe BD is divided into two parts in C, the rectangle DB, BC is equal f to the rectangle BC, CD and the fquare of BC: And f. 3. 2. the doubles of thefe are equal: Therefore the fquares of AB, BC are equal to the fquare of AC, and twice the rectangle DB, BC: Therefore the fquare of AC alone is lefs than the fquares of AB, BC by twice the rectangle DB, BC.
Laftly, Let the fide AC be perpendicular to BC; then is BC the ftraight line between the perpendicular and the acute angle at B; and it is manifeft that the fquares of AB, BC are equal to the fquare of AC, and twice the fquare of BC: Therefore, in every triangle, &c. Q.E. D.
C. 47. I.
PROP. XIV. PROB.
O describe a square that fhall be equal to a given see N' rectilineal figure.
Let A be the given rectilineal figure; it is required to defcribe a fquare that fhall be equal to A.
Defcribe the rectangular parallelogram BCDE equal to the a. 45. 1. rectilineal figure A. If then the fides of it BE, ED are equal
the centre G, at the distance GB, or GF, describe the femicircle BHF, and produce DE to H, and join GH; Therefore, because the ftraight line BF is divided into two equal parts in the point G, and into two unequal at E, the rectangle BE, EF, together with the fquare of EG, is equal to the fquare of b 5.2. GF: But GF is equal to GH; therefore the rectange BE, EF, E
C 47. I.
Book II. together with the fquare of EG, is equal to the fquare of GH: But the fquares of HE, EG are equal to the fquare of GH: Therefore the rectangle BE, EF, together with the square of EG, is equal to the fquares of HE, EG: Take away the fquare of EG, which is common to both; and the remaining rectangle BE, EF is equal to the fquare of EH: But the rectangle contained by BE, EF is the parallelogram BD, because EF is equal to ED; therefore BD is equal to the fquare of EH; but BD is equal to the rectilineal figure A; therefore the rectilineal figure A is equal to the fquare of EH: Wherefore a fquare has been made equal to the given rectilineal figure A, viz. the square described upon EH. Which was to be done.
QUAL circles are those of which the diameters are equal, or from the centers of which the ftraight lines to the circumferences are equal.
'This is not a definition but a theorem, the truth of which ' is evident; for, if the circles be applied to one another, so that ❝their centers coincide, the circles muft likewife coincide, fince 'the straight lines from their centers are equal.'
A ftraight line is said to touch a circle, when it meets the circle, and being produced does not cut it.
Circles are faid to touch one another, which meet, but do not cut one another.
Straight lines are faid to be equally di-
And the ftraight line on which the