Therefore
ADEB is a parallelogram;
therefore AB is equal to DE, and AD to BE. [I. 34]
But
AB is equal to
AD;
therefore the four straight lines BA, AD, DE, EB are equal to one another; therefore the parallelogram
ADEB is equilateral.
I say next that it is also right-angled.
For, since the straight line
AD falls upon the parallels
AB,
DE,
the angles BAD, ADE are equal to two right angles. [I. 29]
But the angle
BAD is right;
therefore the angle ADE is also right.
And in parallelogrammic areas the opposite sides and
angles are equal to one another; [
I. 34]
therefore each of the opposite angles ABE, BED is also right. Therefore ADEB is right-angled.
And it was also proved equilateral.
Therefore it is a square; and it is described on the straight line
AB.
Q. E. F.
1
Proposition 47.
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, it 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]