Advanced High-School Mathematics

22 CHAPTER 1 Advanced Euclidean Geometry

20. The diagram to the right shows
    three circles of different radii with
    centers A, B, and C. The points
    X, Y, and Z are defined by inter-
    sections of the tangents to the cir-
    cles as indicated. Prove thatX, Y,
    andZ are colinear. Z

Y

X

A

B

C

21. (The Euler line.) In this exercise you will be guided through the
    proof that in the triangle 4 ABC, the centroid, circumcenter, and
    orthocenter are all colinear. The line so determined is called the
    Euler line.
    In the figure to the right, let G be the centroid of 4 ABC, and
    letObe the circumcenter. LocateP on the ray

−→

OGso thatGP :

OG= 2 : 1.

(a) Let A′ be the intersection of

(AG) with (BC); show that

4 OGA′ ∼ 4PGA. (Hint: re-

call from page 13 that GA :

GA′= 2 : 1.)

(b) Conclude that (AP) and (OA′)

are parallel which putsPon the

altitude through vertexA.

(c) Similarly, show that P is also

on the altitudes through ver-

ticesBandC, and soP is the

orthocenter of 4 ABC.