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r=\sqrt{\r_1r_2r_3/(r_1+r_2+r_3)}, 3 tangent circles ext. in pairs

Source: 1993 Greece MO p5

September 6, 2024
geometrytangent circles

Problem Statement

Three circles O1, O2, O3O_1, \ O_2, \ O_3 with radiii r1, r2, r3r_1, \ r_2, \ r_3 respectively are tangent extarnally in pairs. Let r be the radius of the inscrined circle of triangle O1O2O3O_1O_2O_3. Prove that r=r1r2r3r1+r2+r3. r=\sqrt{\dfrac{r_1r_2r_3}{r_1+r_2+r_3}}.
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