Projects - Cengage Learning

C-90 Appendix C
avoided in subsequent calculations.) For each exercise, supply the calculations
to show that the given statements are correct.
y
C(0, 6)
A(6a, 0) B(6b, 0)
x
1. The equations of the lines containing the medians of ¢ABC are
2x (a b)y 6(a b), x (a 2b)y 6b, and x (b 2a)y 6a.
2. The intersection point for each pair of medians of ^ABC is (2a 2b, 2).
This point is called the centroid of ^ABC, and it is usually denoted by the
letter G.
3. The equations of the lines containing the altitudes of ^ABC are
y ax 6ab, y bx 6ab, and x 0.
4. The intersection point for each pair of altitudes of ^ABC is (0, 6ab). This
point is called the orthocenter of ^ABC, and it is usually denoted by the
letter H.
5. The equations of the lines containing the perpendicular bisectors of the
sides of ^ABC are x 3a 3b, bx y 3b 2 3, and ax y 3a 2 3.
6. The intersection point for each pair of perpendicular bisectors is
(3a 3b, 3ab 3). This point is called the circumcenter of ^ABC, and it
is usually denoted by the letter O.
7. The orthocenter H, the centroid G, and circumcenter O are collinear. The
line through these three points is called the Euler line of ^ABC.
8. HG 2(GO). That is, the distance from the orthocenter to the centroid is
twice the distance from the centroid to the circumcenter.