EULER’S NINE POINT CIRCLE & LINE CONSTRUCTIONS
You will need a compass and straight edge and five triangles. Mission #5 and #6 are done on the same triangle and only the last triangle needs to be corrected.
MISSION #1 THE CIRCUMCENTER (Use the first triangle)
1) Construct the perpendicular bisector of the 3 sides: KG , GF , FK
2) Label the midpoints M
1
, M
2
, M
3
3) The point where the bisectors meet is the Circumcenter. Label it “C”
4) With center “C” and radius the distance from a vertex, circumscribe the triangle.
5) Go to Mission #2
MISSION #2 THE CENTROID (Use a new triangle)
1) Bisect the sides
2) Label the midpoints M
1
, M
2
, M
3
3) Draw the three medians
4)
5)
The point where the medians meet is the Centroid. Label it “P”.
Go to Mission #3
MISSION #3 THE ORTOCENTER (Use a new triangle)
1)
2)
3)
Construct the altitudes to the three sides:
Label the feet of the altitudes A
1
, A
2
, A
3
KG , GF , FK
The point where the altitudes meet is the Orthocenter. Label it “O”.
4) Find the midpoints of the three segments OK , OF , OG .
5)
6)
Label the midpoints H
Go to mission #4
1
, H
2
, H
3
MISSION #4 THE INCENTER (Use a new triangle)
1) Bisect the angles of the triangle.
2) The point where the bisectors meet is the Incenter. Label it “I”.
3) From the point I, construct a line perpendicular to KF .
4)
Label the foot of the perpendicular “T”.
5) With radius IT, and center I, inscribe a circle in the triangle.
6) Go to Mission #5
MISSION #5 EULER’S NINE POINT CIRCLE (Missions 5 & 6 use the same triangle)
1) Use Mission #1 and a compass to copy the points M
1
, M
2
, M
3
and C onto a new triangle.
2) Use Mission #3 triangle and copy the points A
1
, A
2
, A
3
and H
1
, H
2
, H
3
and O and P onto the new triangle also.
3) Draw the line segment OC
4) Bisect OC and label the midpoint “E”.
5) With center at E and radius EM, draw a circle.
6) How many labeled points are on the circle?
MISSION #6 THE EULER LINE (Use the triangle from Mission #5)
1) Use the Nine-Point Circle drawing. If you do not have nine points on or very close to being on the circle , this construction will not be successful.
2) With a straight edge, draw a line through O and C. If your construction is correct, you should find that points C, P, E, and O are on the line.

NAME____________________________
MISSION NUMBER____________________________
G
K F