By P. K. Jain, Ahmed Khalid
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Extra resources for A Textbook Of Analytical Geometry Of Two Dimensions
If we are given the measure of one angle, for example, ∠A = α, of a triangle ABC and the lengths of the two sides forming that angle, b and c, then we have the following additional formula for the area of triangle ABC. Symbolically that is written as Area ΔABC = bc ⋅ sin ∠A = bc ⋅ sin α. It is also possible to establish the area of a triangle given the lengths a, b, and c, of the three sides of triangle ABC using Heron's formula6 Area ΔABC = , where s = (a + b + c) is the semiperimeter of the triangle ABC.
Using the designations shown in figure 2-7, namely that 42 a1 = BPa , a2 = PaC , b1 = CPb , b2 = PbA , c 1 = APc , c 2 = PcB, we get a12 + b12 + c 12 = a22 + b22 + c 22. INTRODUCTION TO THE MEDIANS OF A TRIANGLE The medians of a triangle, which join a vertex of a triangle with the midpoint of the opposite side, trisect each other at their point of concurrency. That is, in figure 2-8 the following is true: 43 AG = 2 ⋅ GMa BG = 2 ⋅ GMb CG = 2 ⋅ GMc Furthermore, the point G is the center of gravity, or the centroid, of the triangle—that is, the balancing point.
We can see from the fact that each side is divided into two equal segments that the products of the alternate segments will be equal. Therefore, the medians must be concurrent. That can be seen in figure 2-14 (left side), where AMc = McB, BMa = MaC, and CMb = MbA. Since AMc ⋅ BMa ⋅ CMb = McB ⋅ MaC ⋅ MbA, we can conclude (by Ceva's theorem) that the three medians are concurrent. 53 EXPLORING OTHER CONCURRENCIES We are just beginning to scratch the surface of the almostboundless concurrencies that can be found in a triangle.