Question

Given: ΔABC Prove: The three medians of ΔABC intersect at a common point.

Answer 1

Given: ΔABC

When written in the correct order, the two-column proof below describes the statements and justifications for proving the three medians of a triangle all intersect in one point are as follows:

Statements                                                          Justifications

Point F is a midpoint of Line segment AB        by Construction               
Point E is a midpoint of Line segment AC
Draw Line segment BE
Draw Line segment FC 

Point G is the point of intersection between
Line segment BE and Line segment FC               Intersecting Lines Postulate

Draw Line segment AG                                        by Construction

Point D is the point of intersection between
Line segment AG and Line segment BC              Intersecting Lines Postulate

Point H lies on Line segment AG such that
Line segment AG ≅ Line segment GH                 by Construction

Line segment FG is parallel to line segment
BH and Line segment GE is parallel to line
segment HC                                                         Midsegment Theorem

Line segment GC is parallel to line segment
BH and Line segment BG is parallel to
line segment HC                                                  Substitution

BGCH is a                                                        Properties of a Parallelogram parallelogram                                                   (opposite sides are parallel)

Line segment BD ≅ Line segment                    Properties of a Parallelogram DC                                                                    (diagonals bisect each other)   

Line segment AD is a median                          Definition of a Median

Thus the most logical order of statements and justifications is: II, III, IV, I