## Math Forum - Project of the Month

### March and April 1998 Project of the Month

Our guest-problemist this month is Ken Duisenberg. Ken is a Research & Development Engineer with Hewlett-Packard Corporation, creating the newest generation of file-servers for internet applications. He is an avid logic-puzzle enthusiast, with a particular fondness for geometry and math puzzles, and he maintains his own "Puzzle of the Week" site at: http://www.ecst.csuchico.edu/~kend/potw/index.html.

## The Euler Line

1. Define the following terms for a triangle. If you find a good web reference for your definitions, please include the URL in your response.
• a) Centroid
• b) Circumcenter
• c) Orthocenter
• d) Nine-Point Center
• e) Incenter
• f) Euler Line

2. Find as many as you can of (a-f) for the following triangles:
1. Isosceles Right Triangle, with vertices at (0,0), (0,10), (10,0).
2. 30-60-90 Triangle, with vertices at (0,0), (0,10sqrt3), (10,0).
3. 3-4-5 Triangle, with vertices at (0,0), (0,4), (3,0).
4. Equilateral Triangle, with vertices at (-5,0), (0,5sqrt3), (5,0)

3. Create your own triangle, with an Euler Line that does not pass through any of the triangle's vertices, and find as many of (a-f) for your triangle.

4. If you are given the equation for the Euler Line of a triangle, what is the smallest amount of additional information you would need to be able to reconstruct the triangle? For example, if I give you an Euler Line: y=4x/3, what more would you need to build the triangle?
If I add the Centroid: (3,4), is that enough information to make the triangle?
If I add the Circumcenter: (0,0), is that enough?
Can you find a triangle that fits these requirements (give the vertices of your triangle)? Is there only one answer?
Please feel free to add any insights you learned in this study.