Circles and Triangles

Peter Tong

Geometry has been a favorite subject of many of us in our high school days. When some of us came together sometime ago, we tried to remember the definitions of the Appollonius circle and the nine-point circle. The question was which nine points in a triangle lie on a circle. We were not able to find a solution that day and my cursory investigation afterward bore no fruit.

Last weekend, my sister and I went to my parents' house in Philadelphia to prepare the house for sale as my parents had not been living in that house for the last five years. As we went through drawers and drawers of documents and shelves and shelves of books, I came to one with the title A Survey of Geometery by Howard Eves. I had no idea who Eves might be but the table of content looked promising. Instead of throwing it out, I decided to keep it. The book probably belongs to my sister who was a math teacher at one time. Our schools probably no longer teach these subjects. Nevertheless, they could still be interesting to some of us if only for old time sake.

So, here are the definitions as given in the Eves' book.

Appollonius Circle

According to Eves, Appollonius was the last great Greek mathematician in geometry, Euclid being the most famous. The problem of Appollonius is as follow:

Given A and B are fixed points and k a given constant, then the locus of a point P such that AP/BP =k is a straightline if k=1 and is a circle is k11. The circle is known as the Appollonius circle. 

It seems to me that writing a equation for the locus is not that difficult but the trick is to construct this circle using only staight edge and compass. And incidentally, Eves also points out that constructing a circle tangent to three circles is a special case of this problem.

Nine-point Circle

In triangle A1A2A3 let M1,M2,M3 be the mid-points of the sides A2A3, A3A1, A1A2, H1, H2, H3 the feet of the altitudes on these sides, N1, N2, N3 the mid-points of the segments A1H, A2H, A3H, where H is the orthocenter of the triangle. Then, the nine points M1, M2, M3, H1, H2, H3, N1,N2,N3 lie on a circle whose center N is the midpoint of the segment joining the orthocenter H to the circumcenter O of the triangle, and whose radius is half the circumradius of the triangle.

Note: Orthocenter is the point where the three altitudes meet. Circumcircle is the circle passing through the three vertices of the triangle.

As homework, prove these theorems.