Triangle Altitudes and the Circumcircle: an Investigation into Ratios and Relationships

by: Al Byrnes

For this write up, we will first construct an acute triangle ABC:

Next, construct the three altitudes of the triangle which will be denoted by the segments AD, BE, and CF. Recall the definition of an altitude of a triangle: drawing from Weisstein's definition, altitudes (as shown in our illustration below) are the three segments AD, BE, and CF that are perpedicular to the triangle legs AC, BC, and AB and opposite to the verticies B, A, and C, respectivelty. (Weisstein, E. W. Altitude. Retrieved from MathWorld - A Wolf Web Resource.

Weisstein continues, stating that the "three altitudes of any triangle are concurrent at the orthocenter H" (Weisstein, E. W. Altitude. Retrieved from MathWorld - A Wolf Web Resource. This concurrence of the three altitudes is illustrated both above and below, where the illustration features the orthocenter:

Next we will construct the circle, which will circumscribe the triangle ABC, that is the circle that is colinear with each point A, B, and C:

Now, we will extend each altitude to intersect with the circumcircle:

Now that we have created this construction, let's investigate and determine whether there is an interesting relationship between the lengths of the segments formed by the altitudes extended out to the circumcircle and the altitudes themselves (eg: AE, BD, and CF).

In particular, what is sum of the following expression:

My initial conjecture was inspired by some expirimentation with the visual from above. Specifically, what would happen if we were to translate the point D a vector of length an magnitude HD, originating at point D?

After transformation:

Note that D' has ended up at a position that seems to be very close to point on the circumcircle P. While this is not meant to be a convincing proof, I my guess is that the segment HD is equivalent in length to DD' (or rather DP). Using this fact, let's consider the sum of the ratios of chords to altitude lengths:





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