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Vector Theorems - Math Problem Example

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Suppose that ABC is a given triangle as shown in the figure below with vertices given as A= (a1, a2), B= (b1, b2), and C= (c1, c2). Let D, E, and F be the mid-points of sides AB, BC, and AC respectively. By joining these points, a triangle DEF is formed which is called medial triangle.
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Vector Theorems
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Download file to see previous pages Since we have a medial triangle DEF as shown above, for convenience we draw the same triangle separately as shown below. Here D (a, 0, 0), E (0, b, 0), and F (0, 0, c) be the vertices of medial triangle while N is the centroid of the triangle. By distance formula, N has the coordinates as (a/3, b/3, c/3). Let D′ be the mid-point of EF, therefore its coordinates will be (0, b/2, c/2). Now we find the distance between N and D′. Similarly 2 E′N = EN, and 2 F′N = FN. Thus, it means that the centroid is located two thirds of the way from the original vertex to the midpoint of the opposite side of the triangle. Now, since we have a triangle ABC and E is the middle point of BC, and P is the centroid of ABC. Therefore, by using above theorem we have OP = OA + 2/3 AE. Now we will find the centroid of ABC. Now we will find the centroid of DEF. Let Q = centroid of DEF = OQ = OD + 2/3 DM/ From above equation no.2 and equation no.3 it is evident that P=Q. Thus it has been proved that the centroid of the triangle ABC is equal to the centroid of medial triangle DEF ...Download file to see next pagesRead More
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