| The ebook Elementary Calculus is based on material originally written by H.J. Keisler. For more information please read the copyright pages. |
|
Home  Vectors Vectors and Plane Geometry Examples Example 8: A Triangle's Medians Intersecting (Proof) |
|
 Search the VIAS Library | Index
|
  |
Example 8: A Triangle's Medians Intersecting (Proof)
Prove that the lines from the vertices of a triangle ABC to the midpoints of the opposite sides all meet at the single point P given by P = ⅓A + ⅓B + ⅓C.
Figure 10.2.14 PROOF We are given triangle ABC, shown in Figure 10.2.14. Let A', B', C' be the midpoints of the opposite sides. We prove that all three lines AA', BB', CC' pass through the point P. The point A' has position vector A' = ½B + ½C.
The line AA' has the direction vector A' - A. AA' has the vector equation X = A + t(A' - A). The computation below shows that P is on the line AA' p = ⅓A + f⅓B + ⅓C) = ⅓A + ⅔A' = A + ⅔(A' - A). A similar proof shows that P is on BB' and CC.
|
|
| Home Vectors Vectors and Plane Geometry Examples Example 8: A Triangle's Medians Intersecting (Proof) |
|
Last Update: 2006-11-15