26 II. ON TRIANGLES Figure 25 isosceles triangle ACD, ACD = ADC, which is greater than B by the preceding theorem applied to triangle DCB. The theorem is thus proved. Conversely, the greater side corresponds3 to the greater angle. This statement is obviously equivalent to the preceding one. 26. Theorem. Any side of a triangle is less than the sum of the other two. Figure 26 In triangle ABC, we extend side AB to a point D such that AD = AC (Fig. 26). We must prove that4 BC BD. Drawing CD, we see that angle D, which is equal to angle ACD (23), is therefore less than BCD. The desired inequality thus follows from the preceding theorem applied to the triangle BCD. Corollaries. I. Any side of a triangle is greater than the difference between the other two. Indeed, the inequality BC AB + AC gives, after subtracting AC from both sides: BC AC AB. 3 The side corresponding to an angle is the side opposite it. 4 The theorem is obvious if BC is not the largest side of the triangle.
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