Altitude and Orthocenter of a Triangle

An investigation by Ryan Shannon

Given a Triangle ABC

The orthocenter

Let Ortho = E Then Shade Triangle AEB.

As we construct the orthocenter of triangle AEB we notice the orthocenter is point C on triangle ABC

We can do similar construction for triangle BEC

We notice that the orthocenter for triangle BEC is then point A on triangle ABC

Here again find the orthocenter of triangle AEC

We notice the trend as assumption would be that the orthocenter of AEC would be the point B on triangle ABC

After doing this we start to wonder, what would happen should we investigate the circumcenter.

We then find the orthocenter for all the inscribed triangles with point orthocenter to triangle ABC

If we overlap all the circumcircle we begin the think that they maybe the same size.

I encourage you to click on the photo below move vertex ABC to notice how the circumcircle interact.

Lets go further and see what will happen if we join the orthocenter of AEC, BEC, AEB.

SURPRISE!

The orthocenter of the circumcenter is the circumcenter of the given triangle.

Then we ask ourselves is the circumcircle of triangle JKL the orthocenter of the given triangle?

SURPRISE!

The circumcenter of triangle JKL is the orthocenter of the given triangle.

Click on the GSP and notice too, that these circles are equal this the triangles ABC and JKL are similar.