In geometry, the nine-point circle is a circle that can be constructed for any given triangle. [br]This special circle passes through the following points: [br][br]The midpoint of each side of the triangle (D, E, F in applet below)[br]The points at which the lines containing the triangle's altitudes intersect the lines containing the triangle's sides (G, H, I)[br]The midpoint of each segment connecting the triangle's vertex to its orthocenter (J, K, L). [br][br][b]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.) [/b][br][br]And let the fun begin! (See below).

Activity Questions: [br][br]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. [br][br]Use the tools of GeoGebra to do the following. [b] 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)! [/b][br][br]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? [br][br]2) Construct a segment that connects R to O. Prove that R, M, and O are collinear. [br][br]3) For (2) above, how does RM compare to MO? [br][br]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!) [br][br]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. [br][br]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. [br][br]7) Repeat step (6), but this time form quadrilateral LKED. [br][br]8) Repeat step (6) again, but this time form quadrilateral FKJD. [br][br]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?