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confequently the angle ACB is equal to the angle BCD (I. 7.)

But the angle BCD is a right angle (by Conft.), whence the angle ACB is also a right angle.

PROP. XVI. THEOREM.

Q. E. D.

The difference of the fquares of the two fides of any triangle, is equal to the difference of the fquares of the two lines, or diftances, included between the extremes of the base and the perpendicular.

B

Let ABC be a triangle, having CD perpendicular to AB; then will the difference of the fquares of AC, CB be equal to the difference of the fquares of AD, DB.

For the fum of the squares of AD, DC is equal to the fquare of AC (II. 14.); and the fum of the fquares of BD, DC is equal to the fquare of BC (II. 14.)

The difference, therefore, between the fum of the squares of AD, DC and the fum of the fquares of BD, DC, is equal to the difference of the fquares of AC, CB.

And, fince DC is common, the difference between the fum of the squares of AD, DC, and the fum of the squares of BD, DC is equal to the difference of the fquares of ᎪᎠ, ᎠᏴ.

But things which are equal to the fame thing are equal to each other; confequently the difference of the fquares of AC, CB is equal to the difference of the fquares of AD, DB. Q. E. D. COROLL. The rectangle under the fum and difference of the two fides of any triangle, is equal to the rectangle under the base and the difference of the fegments of the base (II. 13.)

PROP. XVII. THEOREM.

In any obtufe-angled triangle, the square of the fide fubtending the obtufe angle, is greater than the fum of the fquares of the other two fides, by twice the rectangle of the base and the distance of the perpendicular from the obtuse angle.

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Let ABC be a triangle, of which ABC is an obtufe angle, and CD perpendicular to AB; then will the fquare of AC be greater than the fquares of AB, BC, by twice the rectangle of AB, BD.

For, fince the right line AD is divided into two parts, in the point B, the fquare of AD is equal to the fquares of

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AE,

AB, BD, together with twice the rectangle of ae, bd (II. 11.)

And if, to each of thefe equals, there be added the fquare of DC, the fquares of AD, DC will be equal to the fquares of AB, BD and DC, together with twice the rectangle of AB, BD.

But the fquares of AD, DC are equal to the square of Ac, and the fquares of BD, DC to the fquare of BC (II. 14.); whence the fquare of AC is greater than the fquares of AB, BC by twice the rectangle of ab, bd. Q. E. D.

PROP. XVIII. THEOREM.

In any triangle, the fquare of the fide subtending an acute angle, is less than the fum of the fquares of the base and the other fide, by twice the rectangle of the base and the distance of the perpendicular from the acute angle.

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Let ABC be a triangle, of which ABC is an acute angle, and CD perpendicular to AB: then will the square of AC, be less than the fum of the fquares of AB and Bc, by twice the rectangle of AB, BD.

For, fince AB, and AB produced, are divided into two parts in the points D, and A, the fum of the fquares of AB,

BD is equal to twice the rectangle of AB, BD, together with the fquare of AD (II. 12.)

And if, to each of these equals, there be added the fquare of DC, the fum of the fquares of AB, BD and DC will be equal to twice the rectangle of AB, BD, together with the fum of the fquares of AD, DC.

But the fum of the fquares of BD, DC is equal to the fquare of BC, and the fum of the fquares of AD, DC to the fquare of AC (II. 14.); whence the fquare of Ac is lefs than the fum of the fquares of AB, BC, by twice the rectangle of AB, BD.

PROP. XIX. THEOREM.

QE. D.

In any triangle, the double of the fquare of a line drawn from the vertex to the middle of the bafe, together with double the fquare of the femi-bafe, is equal to the fum of the fquares of the other two fides.

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Let ABC be a triangle, and ce a line drawn from the vertex to the middle of the bafe AB; then will twice the fum of the squares of CE, EA be equal to the fum of the fquares of AC, cp.

For on AB, produced if neceffary, let fall the perpendicular CD (I. 12.)

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Then, because AEC is an obtufe angle, the fquare of AC is equal to the squares of AE, EC together with twice the rectangle of AE, ED (II. 17.)

And, because BEC is an acute angle, the fquare of CB together with twice the rectangle of BE, ED is equal to the squares of BE, EC (II. 18.)

And fince AE is equal to EB (by Conft.), the fquare of BC together with twice the rectangle of AE, ED is equal to the fquares of AE, EC.

But if equals be added to equals, the wholes will be equal; whence the fquares of AC, CB, together with twice the rectangle of AE, ED, are equal to twice the squares of AE, EC, together with twice the rectangle of AE, ed.

And, if twice the rectangle of AE, EC, which is common, be taken away, the fum of the fquares of AC, CB will be equal to twice the fum of the fquares of AE, EC. Q. E. D.

PROP. XX. THEOREM.

In an ifofceles triangle, the fquare of a line drawn from the vertex to any point in the base, together with the rectangle of the segments of the bafe, is equal to the square of one of the equal fides of the triangle.

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Let ABC be an ifofceles triangle, and CE a line drawn from the vertex to any point in the base AB; then will the

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