Lecture 19
Basic metric relationships in quadrilaterals. The cosines theorem for quadrilaterals.
The straight line segment connecting the midpoints of the opposite sides of a quadrilateral is
called the midline of a quadrilateral.
Theorem. The midlines of a quadrilateral and the straight line segment connecting the
midpoints of its diagonals intersect at one point and are halved at the point of intersection.
C
P
B
E
M
N
G
F
A
D
Q
Proof. Suppose that
and
are the midlines of the quadrilateral
. Let the midpoints
of the diagonals
and
be the points and respectively. Then, according to the properties of
the midline of a triangle the segments
and
are parallel to the diagonal and equal to the half
of it. Therefore, the quadrilateral
is a parallelogram. We can say based on the same property
that the quadrilaterals
and
are also parallelograms. Any two of these three
parallelograms have a common diagonal. The proof of the theorem is clear from the property of the
diagonals of a parallelogram.
Definition. The point of intersection of the midlines of a quadrilateral is called the centroid
of a quadrilateral.
First, let`s make a formula for the dot product of arbitrary
and
vectors.
From the theorem of cosines in the triangles
will be
and
, it
.
Let`s consider these equations in the calculation of the dot product
:
or
.
(1)
Similarly, we can write the equation
.
It turns out that
,
.
Thus, the two opposite sides of a quadrilateral can be perpendicular only if the sum of the squares
of the other two opposite sides is equal to the sum of the squares of the diagonals. The diagonals of a
quadrilateral can be perpendicular only if the sum of the squares of the opposite sides are equal.
Let`s make the formulas for the length of the midline of a quadrilateral and the distance
between the midpoints of its diagonals.
,
,
,
. If we add the equations
side by side and consider
and
we get:
.
If we square both sides of this equation and consider the correlation (1), we get:
Thus, the length of the midline of the quadrilateral is found by the formula
.
Similarly, we can find the distance
:
(2)
between the midpoints of the diagonals of a quadrilateral
.
This relationship is called the Euler`s formula for quadrilaterals.
Let`s review the following particular case.
If the quadrilateral
is parallelogram, the midpoints
and of its diagonals overlap, and from the correlation (3) we
get the formula
.
If
,
,
,
in the quadrilateral
,
The formula
(4)
is correct. This relationship is called the cosine theorem for
quadrilaterals.
,
and the midpoints of the diagonals of
the quadrilateral
and the midpoint of the side are the vertices of the triangle
theorem of cosines in the triangle
, it will be
(3)
. From the
(5)
Considering the Euler`s formula
for quadrilaterals and
Relationships in (5), we get:
Let`s write the values of
cosines in the triangles
and
and
:
.
(6)
from the theorem of
Considering these correlations in (6), we get the formula
(4):
.