PRO P. XV. THEOREM.

If the fquare of one of the fides of a triangle be equal to the fum of the fquares of the other two fides, the angle contained by thofe fides will be a right angle.

Let ABC be a triangle; then if the fquare of the fide AB be equal to the fum of the fquares of AC, CB, the angle ACB will be a right angle.

For, at the point c, make CD at right angles to CB (I. 11.), and equal to AC (I. 3.); and join DB.

Then, fince the fquares of equal lines are equal (II. 2.), the fquare of DC will be equal to the square of AC.

And, if, to each of thefe equals, there be added the fquare of CB, the fquares of DC, CB will be equal to the fquares of AC, CB.

But the fquares of DC, CB are equal to the fquare of BD (II. 14.), and the squares of AC, CB to the fquare of AB (by Hyp.); whence the fquare of BD is equal to the fquare of AB.

And fince equal fquares have equal fides (II. 3.), AB is equal to BD; BC is alfo common to each of the triangles ABC, DBC, and AC is equal to CD (by Conft.);

con

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 alfo a right angle.

PRO P. 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 distances, included between the extremes of the bafe and the perpendicular.

44

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 squares of BD, DC is equal to the fquare of BC (II. 14.)

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

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

AD, DB.

ᎪᎠ, ᎠᏴ.

But things which are equal to the same thing are equal to each other; confequently the difference of the fquares of AC, CB is equal to the difference of the fquares of 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 segments of the bafe (II. 13.)

PROP. XVII. THEOREM.

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

Д

Let ABC be a triangle, of which ABC is an obtufe angle, and CD perpendicular to AB; then will the square 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

...

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

And if, to each of these equals, there be added the fquare of DC, the squares 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 fquare of AC, and the squares of BD, DC to the fquare of BC (II. 14.); whence the fquare of AC is greater than the fsquares of AB, BC by twice the rectangle of AB, BD.

Q.E.D.

PRO P. XVIII.

THEOREM.

In

any triangle, the square of the fide fubtending an acute angle, is lefs than the fum of the squares of the base and the other side, by twice the rectangle of the base and the distance of the perpendicular from the acute angle.

AA

D

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

For, fince AB, or AB produced, is divided into twa parts in the points D, or 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 squares 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. Q. E.D.

PRO P. XIX. THEOREM.

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

4 A

E D

DE

A

Let ABC be a triangle, and CE a line drawn from the vertex to the middle of the base AB: then will twice the fum of the squares of CE, EA be equal to the fum of the fquares of AC, CB.

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