51. Theorem. If, starting at each point of a ﬁgure, we draw equal parallel
segments in the same direction, then their endpoints will form a ﬁgure congruent
to the ﬁrst.
First consider two points A, B of the ﬁrst ﬁgure, and the corresponding points
A , B of the second. Since the segments AA and BB are parallel and equal,
B is a parallelogram. Therefore A B is equal and parallel to AB, with the
same orientation. Thus segments joining homologous pairs of points are equal,
parallel, and have the same direction.
It follows that any three points of the ﬁrst ﬁgure correspond to three points
forming a congruent triangle, and since the angles of these triangles have their sides
parallel and in the same sense, their sense of rotation is the same. The ﬁgures are
The operation by which we pass from the ﬁrst ﬁgure to the second is called a
translation. We note that a translation is determined if we are given the length,
direction, and sense of a segment, such as AA , joining a point to its homologue. We
can therefore designate a translation by the letters of such a segment: for example,
we speak of the translation AA .
Corollaries. I. If, through each point on a line, we draw equal parallel seg-
ments in the same sense, the locus of their endpoints is a line parallel to the ﬁrst.
In particular, the locus of points on the same side of a line, and a at a given
distance from the line, is a parallel line.
II. Two parallel lines are everywhere equidistant.
We can therefore speak of the distance between two parallel lines.
III. The locus of points equidistant from two parallel lines is a third line, parallel
to the ﬁrst two.
Exercise 26. The angle bisectors of a parallelogram form a rectangle. The
bisectors of the exterior angles also form a rectangle.
Exercise 27. Any line passing through the intersection of the diagonals of a
parallelogram is divided by this point, and by two opposite sides, into two equal
For this reason, the point of intersection of the diagonals of a parallelogram is
called the center of this polygon.
Exercise 28. Two parallelograms, one of which is inscribed in the other (that
is, the vertices of the second are on the sides of the ﬁrst) must have the same center.
Exercise 29. An angle of a triangle is acute, right, or obtuse, according as
whether its opposite side is less than, equal to, or greater than double the corre-
Exercise 30. If, in a right triangle, one of the acute angles is double the other,
then one of the sides of the right angle is half the hypotenuse.