Altitudes and Orthocenters


Perform the following constructions

a. Construct any triangle ABC. 

b. Construct the Orthocenter H of triangle ABC.

c. Construct the Orthocenter of triangle HBC.

d. Construct the Orthocenter of triangle HAB.

e. Construct the Orthocenter of triangle HAC.

f. Construct the Circumcircles of triangles ABC, HBC, HAB, and HAC.


After performing these directions I got this construction:

I then constructed the nine point circles for triangles ABC, HBC, HAC, and HAB, found something very interestingÉ


But why is this??


Recall that a nine-point circle passes through nine points: the three midpoints of the sides, the three feet of the altitudes, and the three midpoints of the segments from the respective vertices to the orthocenter. The center of a nine-point circle is the midpoint between the orthocenter and the circumcenter.


The center of this nine-point triangle also lies on the Euler Line. This Euler Line passes through the orthocenter, circumcenter, and centroid of the triangle. Since triangles HBC, HAC, and HAB are all within ABC connecting to its orthocenter, these triangles all have the same nine-point circles.



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