Altitudes and Orthocenters

Kasey Nored

A write-up of an exploration of Altitudes and Orthocenters

We constructed a triangle ABC with an orthocenter H, AHB, BHC, and AHC. The image below clearly shows the three new triangles with a vertex of H.

We constructed the orthocenters of triangles AHB, BHC, and AHC and the circumcenters of each. As shown below.

To further explore we slide a vertex of ABC to H. Our circumcircles collapses into three, which seems reasonable. Shown below.

These three circles all have equal radii and that distance is equal to the medial distance of the triangle ABC, which as you recall is our original triangle with a vertex transfered to H, the orthocenter of our original triangle.

If we label the points where the circles we used to show our equal distance, intersect with the circumcirlces of AHB and CHB, E and F respectively and create segments between the centers of the circles a parallelogram is formed. Here the parallelogram is a rectangle.

Regardless of where we shift the vertex of the triangle ABC we have a parallelogram.

Another parallelogram.

If we remove the circles, the image is clearer.

The parallelogram is only created when ABC is an acute or right triangle.

When ABC is obtuse the orthocenter H, is not inside the triangle.