Exploring the Orthic Triangle

by

Julie Anne Laycock

Take any acute triangle. Construct a triangle connecting the feet of the altitudes. This is called the ORTHIC triangle. Construct G, H, C, and I for the orthic triangle. Compare to G, H, C, and I for the original triangle. Can you extend this for right or obtuse triangles?

Constructing the Orthic triangle.

Step1: Start with an acute triangle.

Step 2: Construct the Altitudes.

Step 3: Mark the feet of the altitudes.

Step 4: Connect the feet of the altitudes.

Construct and compare G, H, C, I for the orthic triangle and the original triangle.

Putting the original triangle with the centers over the orthic triangle and there doesn't appear to be much of a connection between the points.

After changing the angles of the triangle it appears that G, H, I and C for both triangles begin to become the same point as the triangle gets closer and closer to being the medial triangle. It also appears that all the points become collinear as we continue to manipulate the triangle.

Lets look at what happens to the orthic triangle if we have a right triangle or an obtuse triangle.

As we drag the point and the original triangle gets closer and closer to a right triangle we can see the orthic triangle begins to disappear. We create the orthic triangle by connecting the feet of the altitudes. When two of the altitudes become the sides of the original triangle the orthic triangle disappears. When the original triangle is obtuse and two of the altitudes lie outside the orthic triangle does not appear. It will appear when the triangle is acute.