In the $\Delta ABC$, let $r_a,r_b,r_c$ be the exradii, $S$ the area of the triangle, $a,b,c$ its sides, $p$ the semiperimeter and $r$ the inradius. Show that the following inequality holds: $$r_ar_b+r_br_c+r_cr_a \ge 2\sqrt3 S+\frac {abc}{p}+r^2.$$ I tried to express the exradii in terms of $p,a,b,c$, but I didn't get to show that the inequality holds.
2026-03-27 06:56:55.1774594615
Inequality involving inradius, exradii, sides, area, semiperimeter
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