2016 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 10

Problem

Let $P$ be a point on the triangle $\triangle ABC$ (inside or on the boundary). Let $r_a , r_b$ and $r_c$ be the distance from $P$ to the sides $BC$ , $CA$ and $AB$ , respectively.

Show that

a) $r_a\cdot{a} +r_b\cdot{b} <=|PC|\cdot{c}$ and also $r_a\cdot b + r_b\cdot a<=|PC|\cdot c$ , where $a = |BC|, b = |CA|$ and $c = |AB|$ .

b) (Assuming the inequalities of part a) $\frac{|PA| + |PB| + |PC|}{r_a + r_b + r_c}>=2$ .

2016 UNM-PNM Contest II (Problems • Answer Key • Resources)
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2016 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 10

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