50 VI. ON PARALLELOGRAMS. — ON TRANSLATIONS

Figure 48

49. A square is a quadrilateral in which all sides are equal, and all the angles

right angles.

Thus a square is both a rhombus and a rectangle, so that the diagonals are

equal, perpendicular, and intersect each other at their common midpoint.

Conversely, any quadrilateral whose diagonals are equal, perpendicular, and

bisect each other is a square.

Two squares with the same side are congruent.

50. Translations.

Lemma. Two ﬁgures F , F are congruent, with the same sense of rotation, if

their points correspond in such a way that, if we take points A, B, C from one

ﬁgure, and the corresponding points A , B , C from the other, then the triangles

thus formed are always congruent, with the same sense of rotation, no matter which

point we take for C.

Indeed, take two points A, B in ﬁgure F , and the

homologous2

points A , B .

Segment AB is then clearly equal to A B . Let us move the second ﬁgure onto the

ﬁrst in such a way that these two equal segments coincide. We claim that then

the two ﬁgures coincide completely. Indeed, let C be a third point of the ﬁrst

ﬁgure, and let C be its homologous point. Since the two triangles ABC, A B C

are congruent, angle B A C is equal to BAC, and has the same sense of rotation.

Therefore, when A B is made to coincide with AB, the line A C will assume the

direction of AC. Since A C = AC as well, we conclude that C coincides with

C. This argument applies to all the points of the ﬁgure, so the two ﬁgures must

coincide completely.

Remarks. I. We have just provided a suﬃcient condition for two ﬁgures to be

congruent; this conditions is also clearly necessary.

From the preceding reasoning it also follows that:

II. In order to superimpose two equal ﬁgures with the same orientation, it

suﬃces to superimpose two points of one of the ﬁgures onto their homologous

points.

2This

is the name given to pairs of corresponding points in the two ﬁgures.