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2/(1/d_a + 1/d_b + 1/dc) < r < (d_a + d_b + d_c)/2

Source: 1985 Polish MO Finals p4

January 21, 2020
geometryinradiusinequalitiesGeometric Inequalities

Problem Statement

PP is a point inside the triangle ABCABC is a triangle. The distance of PP from the lines BC,CA,ABBC, CA, AB is da,db,dcd_a, d_b, d_c respectively. If rr is the inradius, show that 21da+1db+1dc<r<da+db+dc2\frac{2}{ \frac{1}{d_a} + \frac{1}{d_b} + \frac{1}{d_c}} < r < \frac{d_a + d_b + d_c}{2}
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