# A Nine Point Circle Exploration

Topic:
Circle

## Background

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.

## 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!

## 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.