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 land 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 land m which are not parallel to each other, but such that they are both parallel to a third line n . Thus, lines land m intersect at some point. Let T be this point where lines land m intersect . (See the figure.) Since land m are both parallel to line n , point T is not on line n . Thus, lines land 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 land n are both parallel to m , and so by the "Common Parallel" Theorem, Theorem 4.2.9, l|| n. lmnT
Recall the "Opposite Sides of a Parallelogram" Theorem, Theorem 4.2.4, proven earlier, states that the opposites sides of a parallelogram are congruent. Definition (NIB): Three or more mutually parallel lines are equally spaced linesif there is a common transversal which is cut into mutually congruent segments (by them). The next theorem states that every line intersecting three equally spaced lines are cut by them into mutually congruent segments.