For a like reason ACH is a straight line.

L

and FAB CAD, each being a L+b;

▲ ABF: ▲ ACD.

(Cons.)

(83)

Square AG and ▲ ABF have a common base AF and a

255. COR. 1.-The square on either side about a right-angled triangle is equivalent to the square on the hypothenuse minus the square on the other side.

256. COR. 2.-The square on the diagonal of a square is double the given square.

257. COR. 3.-The diagonal and the side of a square are incommensurable.

Let d be the diagonal, and a the side of a square.

d

a

a

PROJECTION.

DEFINITIONS.

258. The Projection of a Point

upon an indefinite

straight line is the foot of the perpendicular drawn from the point to the line.

Thus, the projection of the point C upon the line AB is the point E.

The Projection of a Finite Straight Line upon an indefinite one, is the part of the line intercepted between the perpendiculars drawn from the extremities of the finite line. Thus, EF is the projection of CD upon AB.

If one extremity of CD rests upon the other line AB, then

the projection of CD is ED.

THEOREM VII.

259. In any triangle, the square on the side opposite an acute angle equals the sum of the squares of the other two sides minus twice the product of one of those sides and the projection of the other upon that side.

In the

ABC, let c be an acute , and PC the projection of AC upon BC.

then AP2 + PB2 = BC2 + AP2 + PC2 — 2 BC × PC.

THEOREM VIII.

260. In any obtuse-angled triangle, the square on the side opposite the obtuse angle equals the sum of the squares of the other two sides plus twice the product of one of those sides and the projection of the other upon that side.

In the ▲ ABC, let c be the obtuse ▲, and PC the projection of AC upon BC produced.

Then AP+PB' = BC2 + AP2 + PC2 + 2 BC × PC.

THEOREM IX.

261. In any triangle, if a medial line is drawn to the base:

I.—The sum of the squares of the two sides equals twice the square of half the base plus twice the square of the medial line.

II. The difference of the squares of the two sides equals twice the product of the base and the projection of the medial line upon the base.

In the ▲ ABC, let AD be the medial line, and DP the projection of AD upon the base BC.

I-That AB+ AC2 BD2 + 2 AD'.

2

II. That AB A C2

=

2 BC X DP.

If AB > AC, a is obtuse, and b is acute.

=

Add these equations, observing that BD DC.

Then A B2 + A C2 = 2 B D2 + 2 A D2.

Subtract the second equation from the first.

Then AB AC 2 BCX DP.

=

(260)

(259)

Q. E. D.