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.
3The
side corresponding to an angle is the side opposite it.
4The
theorem is obvious if BC is not the largest side of the triangle.
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