VI. ON PARALLELOGRAMS. ON TRANSLATIONS 49
Remark. We have proved the converses in 46 and 47 by retracing the original
reasoning in the opposite direction, as explained in 32, Remark II.
48. A quadrilateral whose angles are all equal, and are therefore are all right
angles, is called a rectangle. A rectangle is a parallelogram, since its opposite angles
are equal.
A quadrilateral whose sides are all equal is called a rhombus. A rhombus is a
parallelogram since it has equal opposite sides.
Thus, in a rectangle, as in a rhombus, the diagonals intersect at their common
midpoint.
Theorem. The diagonals of a rectangle are equal.
In rectangle ABCD (Fig. 47), the diagonals are equal because triangles ACD,
BCD are congruent: they have side DC in common, ADC = DCB since they are
both right angles, and AD = BC since they are opposite sides of a parallelogram.
Corollary. In a right triangle, the median from the vertex of the right angle
equals half the hypotenuse.
This is true because if we draw parallels to the sides of the right angle through
the endpoints of the hypotenuse, we form a rectangle, in which the median in
question is half the diagonal.
Converse. A parallelogram with equal diagonals is a rectangle.
Figure 47
Suppose (Fig. 47) that AD = BC in parallelogram ABCD. We know that
AD = BC: consequently, triangles ADC, BCD are congruent, since their three
sides are equal in pairs. Angles ADC, BCD are therefore equal and, since they
are supplementary, they must be right angles, which shows that the parallelogram
is a rectangle.
Corollary. A triangle in which a median is one half the corresponding side
is a right triangle.
Corollary. In a rhombus, the diagonals are perpendicular, and bisect the
vertex angles.
If ABCD (Fig. 48) is a rhombus, then triangle ABD is isosceles. Diagonal
AC, being a median of this triangle, is also an altitude and an angle bisector.
Converse. A parallelogram with perpendicular diagonals is a rhombus.
Indeed, each vertex is equidistant from the adjacent vertices, since it lies on
the perpendicular bisector of the diagonal joining these vertices.
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