Projects - Cengage Learning

Appendix C C-89
2. An altitude of a triangle is a line segment drawn from a vertex perpendicular
to the opposite side. A theorem from geometry states that in any triangle
the three altitudes are concurrent. The point where they meet is called
the orthocenter of the triangle. Follow Exercise 1 using the altitudes of
^ABC, rather than the medians.
3. A theorem from geometry states that in any triangle the perpendicular bisectors
of the three sides are concurrent. The point where they intersect is
called the circumcenter of the triangle. The circumcenter is the center of
the circle that passes through the three vertices of the triangle. Follow
Exercise 1 using the perpendicular bisectors of the sides of ^ABC rather
than the medians.
4. In Exercises 1–3 you found the coordinates of the centroid, the orthocenter,
and the circumcenter of ^ABC.
(a) Show that these three points are collinear (that is, all lie on a straight
line). Hint: Use slopes.
(b) Show that the distance from the orthocenter to the centroid is twice the
distance from the centroid to the circumcenter.
Remark: It can be shown (Workbook 2) that the results in parts (a) and (b)
are valid for all triangles. The line containing the centroid, orthocenter, and
circumcenter is called the Euler line of the triangle.
5. (a) Find the center and radius of the circle that passes through the midpoints
of the sides of ^ABC. Hint: Find the center as the intersection
of two perpendicular bisectors, as in Exercise 3.
(b) Show that the center lies on the Euler line of ^ABC and that it is situated
halfway between the orthocenter and circumcenter of ^ABC.
6. Let H denote the orthocenter of ^ABC (determined in Exercise 2). Find the
midpoints of the segments HA, HB, and HC. Then check that each of these
midpoints lies on the circle determined in Exercise 5.
7. In ¢ABC, find the foot of each altitude (that is, the point where the altitude
intersects the opposite side.) Then check that each of these three points lies
on the circle determined in Exercise 5.
Remark: In view of Exercises 5–7, the following nine points all lie on the
same circle: the feet of the three medians, the feet of the three altitudes, and
the midpoints of the three segments drawn from the orthocenter to each
vertex. It can be shown that this occurs no matter what triangle we begin
with. The circle is called the nine-point circle.
8. Use a graphing utility to display ^ABC, the nine-point circle, and the
nine special points that all lie on the circle.
Workbook 2
For this workbook you are given an arbitrary triangle with vertices as indicated
in the following figure. (All triangles can be represented in this fashion:
The units on the y-axis are chosen so that the length of that altitude is 6; the
constant a can be any negative real number, while b can be any positive real
number. The factor 6 has been chosen so that the appearance of fractions is