Nine Point Circle
Nine Point Circle: Discovery credited to Olry Terquem, Jean-Victor Poncelet, and Karl Feurbach
Step 1: Construct a triangle using the polygon tool, then find the midpoints of its sides. Those midpoints are the first three of the nine points.
Step 2: Construct three perpendicular lines from the vertices of the triangle through the opposite sides to find the altitudes. Where the altitudes intersect is the orthocenter. The intersection points of the altitudes and sides are the next three of the nine points. Hide the altitudes using the algebra bar.
Step 3: Construct line segments between the orthocenter and the vertices of the triangle and find the midpoints: these are the remaining three points. Hide the line segments.
Step 4: Construct perpendicular bisectors of the sides of the original triangle to find the circumcenter, then find the midpoint of the circumcenter and the orthocenter. Hide the bisector.
Step 5: Construct a circle with that midpoint as the center. It will go through all nine points.
Congratulations, you have a nine point circle!
Want a challenge? See what happens if you make the triangle right!