A Nine Point Circle Exploration



Over the next few days, we are going to explore a famous geometric shape: the nine point circle. The nine point circle gets its name from the fact that we construct it from the nine important points of a triangle. Each triangle has its own nine point circle. To begin, we're going to do some quick definitions.

What's an altitude?

Check for Understanding

So... what are the altitude and feet of a triangle?

Step One: Build your Nine Point Circle.

Now it's time to create a nine point circle. Once you do, we'll be doing several things with it. Here are the instructions: 1. Construct triangle ABC 2. Find the midpoint of AB, BC, and CA and label them as L, M, and N respectively. 3. Create the altitude from each side of your triangle. Label the foot of the altitude from C to AB as D, from A to BC as E, and from B to CA as F. 4. The altitudes should be concurrent at a point. Label that point H. 5. Find the midpoint of AH, BH, and CH. Label these points X, Y, and Z respectively. 6. You can now construct the circle through L, M, N, D, E, F, X, Y, Z and you have made a nine point circle!

The finished product

Some Review

What is the incenter of a triangle?

Some Review

What is the circumcenter of a triangle?

Some Review

What does it mean to circumscribe a triangle?

Some Review

What does it mean to inscribe a triangle?

Some (Challenging) Review

What does it mean if a line is tangent to a circle?

So now what?

Now that you've seen one, it is time for you to construct a nine point circle yourself. Give it a shot.