Geometry: An Interactive Journey to Mastery

6. Points X, Y, and Z are midpoints of the line segments on
   which they lie. Explain why ++XYZ~.ABC What is the
   VFDOHIDFWRU"ͼ௘6HHFigure 13.13௘ͽ

Midpoints have been the focus of this lesson. But there is actually
nothing special about them. Analogous results apply for other
types of points as well. The next three problems demonstrate this.

7. In +ABC, the points M and N lie^58 of the way down the sides on which
   WKH\OLHͼ௘6HHFigure 13.14௘ͽ
   Prove:
   D௘ͽ MN is parallel to the base BC.
   E௘ͽ /HQJWKMN is^58 the length of BC.


8. Consider a set of triangles sharing a common line segment as
   LWVEDVHͼ௘6HHFigure 13.15௘ͽ:KDWFDQ\RXVD\DERXWDOOWKH
   line segments connecting the points^58 of the way down the
   sides of these triangles?


9. Explain why segments connecting the points^58 of the way down
   two pairs of sides of a quadrilateral, as shown in Figure 13.16,
   must be parallel.

B

A C

X O

Y

Z

Figure 13.13

B

M

A

N

C

Figure 13.14

Figure 13.15

Figure 13.16