EMAT 6700

Nine-Point Circle

Use geometer's sketchpad to construct the nine-point circle of a triangle.

Construct the nine-point circle for triangle ABC.

1.  Three points of the nine-point circle are the midpoints K, L, and M of the sides of triangle ABC.

2.  The next three points of the nine-point circle are the intersections of the three altitudes and their respective opposite  sides of triangle ABC.  Hence, the point X, Y, and Z, where segment AX, segment BY, and segment CZ are the altitudes of triangle ABC.

3.  Point H is the intersection of the three altitudes.  It is important because it enables the segments AH, BH, and CH to be constructed.

4.  The points R, S, and T which are the midpoints of the segments AH, BH, and CH respectively are the final three points of the nine points on the nine-point circle of triangle ABC.

5.  Since angle SYL is a right angle, then segment SL is a diameter of the nine-point circle.  Likewise, angles TZM and RXK are right angles.  Therefore, segments TM and RK are also diameters of the nine-point circle.  The point that all the diameters of a circle have in common is the center of the circle.  Hence, point O is the center of the nine-point circle of triangle ABC.

6.  Select point O and point L.  Construct segment OL.  Select point O and segment OL.  Construct circle by center and radius.  Select segments OL, SL, RK, and TM.  Hide the segments.

Circle O is the nine-point circle for triangle ABC.