Advanced High-School Mathematics

44 CHAPTER 1 Advanced Euclidean Geometry

we need only recall that not all quadrilaterals are cyclic. Yet, as we

see, if the nine points are carefully—but naturally—defined, then such

a circle does exist!

Theorem. Given the triangle 4 ABC, construct the following nine

points:

(i) The bases of the three altitudes;

(ii) The midpoints of the three sides;

(iii) The midpoints of the segments join-
ing the orthocenter to each of the
vertices.

Then there is a unique circle passing through these nine points.

Proof. Refer to the picture below, whereA, B,andCare the vertices,

andX, Y, andZare the midpoints. The midpoints referred to in (iii)

above areP, Q, andR. Finally,Ois the orthocenter of 4 ABC.

C

X

B

Q

Z' Z

A

R

O

P

Y'

Y

X'

By the Midpoint Theorem (Exercise 3 on page 6 applied to 4 ACO, the

line (Y P) is parallel to (AX′). Similarly, the line (Y Z) is parallel to

(BC). This implies immediately that∠PY Zis a right angle. SImilarly,

the Midpoint Theorem applied to 4 ABC and to 4 CBOimplies that