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4

Part of 2008 Dutch Mathematical Olympiad

Problems(1)

radius of circle ext. tangent to 3 circles, externally tangent also

Source: Dutch NMO 2008 p4

8/28/2019
Three circles C1,C2,C3C_1,C_2,C_3, with radii 1,2,31, 2, 3 respectively, are externally tangent. In the area enclosed by these circles, there is a circle C4C_4 which is externally tangent to all three circles. Find the radius of C4C_4.
[asy] unitsize(0.4 cm);
pair[] O; real[] r;
O[1] = (-12/5,16/5); r[1] = 1; O[2] = (0,5); r[2] = 2; O[3] = (0,0); r[3] = 3; O[4] = (-132/115, 351/115); r[4] = 6/23;
draw(Circle(O[1],r[1])); draw(Circle(O[2],r[2])); draw(Circle(O[3],r[3])); draw(Circle(O[4],r[4]));
label("C1C_1", O[1]); label("C2C_2", O[2]); label("C3C_3", O[3]); [/asy]
geometrytangent circlesradiusDutch MOexternal tangent