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First we will construct the orthocenter of triangle ABC. In order to do this we will have to construct all three altitudes of triangle ABC. We know by definition that the intersection point of all of these altitudes will be the orthocenter of triangle ABC. We will call the orthocenter point H.

Next, we will construct each of the points where the altitude intersects the opposite side of triangle ABC. We will label these points D, E, F. Points D, E, and F will be the feet of the perpendiculars from A, B, and C respectively.

 

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First we are going to prove the following:

 

Let's consider the following:

Now consider the ratios of areas of the triangles.

Furthermore,

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Lastly, we will prove the following:

We can rewrite these segments by using segment addition properties.

Therefore we know that

AH = AD - HD

BH = BE - HE

CH = CF - HF

Now observe as we substitute these values into our previous equation.

Therefore,

.

Surprise our proof is done!

 

Take Note: we used the previous result we proved above,

.

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Recall that we proved the conditions above for acute triangles only. An obtuse triangle has one angle that has a measure greater than 90 degrees. Are these two properties valid if we have an obtuse triangle.

For the obtuse triangle above we see that HD > AD so

.

This shows that this property does not hold for obtuse triangles.

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Using a similar analysis as above we can show that

.

 

 

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These conditions will only hold true if the triangle is an acute triangle. Explore this GSP construction, Altitudes and the Orthocenter, to see what happens if the triangle is obtuse. It's quite interesting.

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