Assignment #8

Altitudes and Orthocenters

By

Michelle Nichols

Construct any triangle ABC.

Construct the Orthocenter H of triangle ABC.

Construct segments between H and all vertices of triangle ABC, to get Triangle HBC, HAB, and HAC.

If we construct the orthocenters of each of these triangles, each will have an orthocenter the same as a vertice of Triangle ABC.

For Triangle HBC, the Orthocenter is Point A

For Triangle HAB, the Orthocenter is Point C

For Triangle HAC, the Orthocenter is Point B

Now, construct the circumcircles for all triangles: ABC, HBC, HAB, and HAC.

What if we look at the nine-point circle?

Triangles ABC, HBC, HAB, and HAC all share the same Nine Point Circle! Click here to observe this sketch in GSP.

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