3.2 Parallel Lines¶
We know the medians of a triangle are concurrent at a point called the centroid, but proving it requires a result about sets of equally-spaced parallel lines.
- Median
A line passing through a vertex and the midpoint of the opposite side.
Fig. 9 Demonstrating the three median segments of a triangle.¶
A Theorem for the Toolbox
The Transversal Congruence theorem is used only once, but we need it to prove that the medians of a triangle are concurrent.
Transversal Congruence Lemma. If a transversal intersects three parallel lines in such a way as to make congruent segments between the parallels, then every transversal interesecting these parallel lines will do likewise.
Fig. 10 A sketch showing Transversal Segment Congruence with a hint for the proof.¶
Theorems¶
Transversal Congruence Theorem. 1 If a transversal intersects \(n\) parallel lines (\(n>2\)) in such a way as to make congruent segments between the parallels, then every transversal interesecting these parallel lines will do likewise.
Median Concurrence Theorem. The three medians of a triangle are concurent. The concurrency point, called the centroid, is two-thirds of the distant from any vertex to the midpoint of the opposite side.
Hint
Use the Transversal Segment Congruence Theorem to prove the Median Concurrence Theorem.
Two lines parallel to another line are parallel to each other.
If a line intersects one of two parallel lines, then it interesects the other.
Parallel lines are everywhere equidistant.
Warning
Parallel lines are everywhere equidistant only in Euclidean geometries. This visualization is often taught to school children as a definition of parallel, but in formal geometry, two lines being parallel simply means that they do not intersect.
- 1
This theorem name is non-standard, invented and used in this document only.