Triangles Formed During the Construction of the Orthocenter
By: Lacy Gainey
Given acute triangle ABC. Construct the Orthocenter H. Let points D, E, and F be the feet of the perpendiculars from A, B, and C respectfully.
First, we will
Proof:
Let the Area of ΔABC = K. Using the formula Area = ½(base)(height), we know
The Area of ΔABC, K, is also equal to the sum of the 6 smaller triangles formed during the construction of the orthocenter.
Observe,
Now we will
Proof:
We now know that
are illustrated in this GSP file.
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Do these proofs hold when ΔABC is obtuse?
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Drag points A, B, & C and observe what happens when ΔABC is obtuse.
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When ΔABC is obtuse it appears that 2 of the 3 perpendicular segments located within our triangle disappear, in addition to the orthocenter.
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In reality, when ΔABC is not acute, 2 of the 3 perpendicular segments are intersecting 2 of the triangle’s vertices (when ΔABC is a right trianlgs) or are no longer located within the interior of ΔABC (when ΔABC is obtuse). They have not disappeared.
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The orthocenter has not disappeared, but is instead no longer located within the triangles interior.
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Since all of the smaller triangles formed during the construction of the orthocenter are no longer located within the interior of ΔABC, the areas of the smaller triangles do not sum to the are of ΔABC. Therefor, these proofs do not hold when ΔABC is obtuse.