EMAT 6600

Problem Solving

Altitude Theroem ---- Equilateral Triangle

Problem:  Take an equilateral triangle. Pick any point P inside the triangle.

We ought to show that the sum of the segment DP, EP and FP is equivalent to the altitude of the triangle ABC.  There are a few ways to do so.  Here are two of them.

First:  construct the triangles APB, APC and BPC.  What we know is that the sum of the area of these three triangles equals the area of triangle ABC.

Let h1, h2, and h3 be the altitudes of the three triangles respectively; and h be the altitude of the triangle ABC.

Consequently, we have ½ h1b + ½ h2b + ½ h3b = ½ hb.  Notice that since the triangle ABC is an equilateral triangle all the segments of the triangles are congruent.

If we now multiply the equation by 2b, we obtain the following,

h1 + h2 + h3 = h, as required.

Second: This way is rather clever.  However, it requires a bit of familiarity with GSP.

We can translate the points the segments to where they are aligned.  Since, we are dealing with an equilateral triangle we will rotate the points 60 degrees.  Click here to see it.

Now, you can either measure both segments to compare or merge the point P on the altitude of the triangle ABC.

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