Notes on Median Concurrence
Recall that we are working within Euclidean Geometry.
Therefore, the following
statements are all true:
1.
The Euclidean Parallel Postulate is true:
Given any line
l
and any point P not on line
l
, there is one and only one line m which
passes through point P and is parallel to line
l
.
2.
The statement, "Every triangle has angle sum equal to 180
," is true.
3.
The Converse of the Alternate Interior Angle Theorem is true:
If two parallel lines are intersected by a transversal (line), then the alternate interior
angles along the transversal are congruent.
It is also true in Euclidean Geometry (as shown below) that the parallelism of lines is transitive,
which follows from Theorem 4.2.9, which is proved here.
Theorem 4.2.9, The "Common Parallel" Theorem
:
Two lines (which are) parallel to the same line are parallel to each other.
Proof:
Suppose, by way of contradiction, that the statement of the theorem is false.
Then,
there exist two distinct lines
l
and m which are not parallel to each other, but
such that they are both parallel to a third line n .
Thus, lines
l
and
m
intersect at some point.
Let
T
be this point where lines
l
and
m
intersect .
(See the figure.)
Since
l
and
m
are both parallel to line
n ,
point
T is not on line
n .
Thus, lines
l
and
m
are two distinct lines passing
through T and parallel to line
n .
But, by the
Euclidean Parallel Postulate, there is one and only
one line which passes through T and is parallel to
line
n ,
WHICH IS A CONTRADICTION
.
Therefore, the statement of the theorem is true.
Thus, any two lines which are parallel to the
same line are parallel to each other.
QED
Transitivity of parallelism follows because, if
l
|| m
and m ||
n ,
then
l
and
n
are both parallel to
m ,
and
so by the "Common Parallel" Theorem, Theorem 4.2.9,
l
||
n
.
l
m
n
T