9-Point Circle (Informal Investigation)
In geometry, the nine-point circle is a circle that can be constructed for any given triangle.
This special circle passes through the following points:
The midpoint of each side of the triangle (D, E, F in applet below)
The points at which the lines containing the triangle's altitudes intersect the lines containing the triangle's sides (G, H, I)
The midpoint of each segment connecting the triangle's vertex to its orthocenter (J, K, L).
There are many cool features about a triangle's 9-point circle. As you complete the investigation questions below this applet, be sure to continually MOVE VERTICES A, B, & C AROUND to informally validate that any conjecture you make STILL HOLDS TRUE (for many different possible triangles.)
And let the fun begin! (See below).
Activity Questions:
For these questions, we'll denote the circumcenter as "C", the orthocenter as "O", the centroid as "R", and the incenter as "S". Let's denote "M" as the center of the 9-point circle.
Use the tools of GeoGebra to do the following. As you do, be sure to MOVE VERTICES A, B, & C AROUND to informally validate that any conjecture you make STILL HOLDS TRUE (for many different possible triangles)!
1) Construct the triangle's circumcircle and measure its radius. Measure the radius of the 9-pont circle. What is the ratio of the larger radius to the smaller radius?
2) Construct a segment that connects R to O. Prove that R, M, and O are collinear.
3) For (2) above, how does RM compare to MO?
4) Construct any point "W" that lies on the circumcircle you've just constructed in (1) above. Construct a segment connecting O to W. Plot and label a point Y at which this segment intersects the 9-point circle. What seems to be true about OY and YW (regardless of where point W lies?--Try moving it around!)
5) Construct the triangle's Euler Line (line that passes through C, R, and O). Show that M is also collinear with these 3 points. In addition, find & simplify the ratio CM : MO. Also, find and simplify the ratio CM: CO.
6) Construct segments to form the quadrilateral FEJL. What special type of quadrilateral does this polygon look like? Use the tools of GeoGebra to informally prove your conjecture.
7) Repeat step (6), but this time form quadrilateral LKED.
8) Repeat step (6) again, but this time form quadrilateral FKJD.
9) Construct the triangle's incircle (inscribed circle). At how many points do the triangle's incircle and 9-point circle intersect? Where is this point of intersection?