(Edited by Sir Thomas L. Heath, 1908)

[Euclid, ed. Heath, 1908, on

In right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle.

Let ABC be a right-angled triangle
having the angle
BAC right;

I say that the square on BC
is equal to the squares on
BA, AC.

For let there be described on BC
the square BDEC,

and on BA,
AC the squares
GB, HC;
[I. 46]

through A let
AL be drawn parallel to either
BD or CE,
and let AD, FC
be joined.

Then, since each of the angles
BAC, BAG
is right, if follows that with a straight line
BA, and at the point
A on it, the two straight lines
AC, AG
not lying on the same side make the adjacent angles equal to two right
angles;

therefore CA is in a straight line with
AG.
[I. 14]

For the same reason

BA is also in a straight line
with AH.

And, since the angle DBC
is equal to the angle FBA: for
each is right:

let the angle ABC
be added to each;

therefore the whole angle DBA
is equal to the whole angle
FBC.
[C.N. 2]

And, since DB is equal to
BC, and
FB to
BA,

the two sides AB,
BD are equal to the two sides
FB, BC
respectively;

and the angle ABD is equal to
the angle FBC;

therefore the base AD is equal to
the base FC,

and the triangle ABD is equal to
the triangle FBC.
[I. 4]

Now the parallelogram BL
is double of the triangle ABD,
for they have the same base BD
and are in the same parallels
BD, AL.
[I. 41]

And the square GB is double of the
triangle FBC,

for they again have the same base FB
and are in the same parallels.
FB, GC
[I. 41]

[But the doubles of equals are equal to one another.]

Therefore the parallelogram BL
is also equal to the square
GB.

Similarly, if
AE, BK
be joined,

the parallelogram CL can also be
proved equal to the
square HC;

therefore the whole square BDEC
is equal to the two
squares GB, HC.
[C.N. 2]

And the square BDEC
is described on BC,

and the squares
GB, HC
on BA, AC.

Therefore the square on the side BC is equal to the squares on the sides BA, AC.

Therefore etc. Q.E.D.

Book I: Euclid, Book I (ed. Sir Thomas L. Heath 1st Edition, 1908)

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