The Euler Segment of a Triangle In this problem we will demonstrate an intriguing property...

The Euler Segment of a Triangle In this problem we will demonstrate an intriguing property of the important points associated with any triangle: the circumcenter O, the centroid G, and the orthocenter H.

[1] Construct ΔABC.

[2] Construct two perpendicular bisectors of the sides of ΔABC (locate two midpoints of sides using Point At Midpoint from the CONSTRUCT menu, then Perpendicular from that menu). Select the point of intersection F ( = 0), which is the circumcenter of ΔABC, and hide the two lines.

[3] Using the midpoints D and E from Step 2, construct segments from the opposite vertices, and select the point of intersection, G. This is the centroid of ΔABC. Hide the two segments used to find G, and the midpoints D and E.

[4] Construct two altitudes of ΔABC (select A, then segment BC, and choose Perpendicular Line from the CONSTRUCT menu, to get the altitude from A). Select the point H of intersection. This is the orthocenter of ΔABC. Hide the altitudes just constructed.

[5] Construct segment

(a) Does it appear that the points O (= F), G, and H are collinear? To provide an accurate test, calculate the distances FG, GH, and FH, and then the value x = FG + GH. How does x compare with FH? Drag vertex B and observe the effect.

(b) For a further relationship, examine the distances FG and FH. Do you find that FH = 3 • FG? To find out, exhibit the value FH - 3 • FG on the screen. Again drag B to see if this affects that relationship.

(c) State the general theorem you illustrated here.

NOTE: Save this construction for the next problem.