Ratio of Segments from Orthocenter and Perpendiculars

By Joshua Singer

Given triangle ABC, construct the orthocenter H. Let points D, E, and F be the feet of the perpendiculars from A, B, and C respectfully. Prove the following two formulas:

1.

2.

First, let's get a graphical representation of this so that we know what we are proving. Below is triangle ABC with orthocenter H and points D, E, and F defined above.

As you can see, we have divided the triangle into three smaller triangles. We are going to use this to help prove our two formulas. Let's start with the first formula:

To prove this, we are going to use our knowledge of areas of triangles.

Now, let's prove the second formula:

This proof will look very similar to the one we just did, but we will use our knowledge of line segments to expand upon it.

If you notice, the graphical representation that we gave for this proof was an acute triangle. Do the relationships hold true for an obtuse triangle? Click HERE to change the triangle and observe what happens when you have an obtuse triangle. You should notice that when the triangle is obtuse, the relationships do not hold since some of the segments no longer exist.

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