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National and Regional Contests
Greece Contests
Greece National Olympiad
1994 Greece National Olympiad
5
5
Part of
1994 Greece National Olympiad
Problems
(1)
r=\sqrt{\r_1r_2r_3/(r_1+r_2+r_3)}, 3 tangent circles ext. in pairs
Source: 1993 Greece MO p5
9/6/2024
Three circles
O
1
,
O
2
,
O
3
O_1, \ O_2, \ O_3
O
1
,
O
2
,
O
3
with radiii
r
1
,
r
2
,
r
3
r_1, \ r_2, \ r_3
r
1
,
r
2
,
r
3
respectively are tangent extarnally in pairs. Let r be the radius of the inscrined circle of triangle
O
1
O
2
O
3
O_1O_2O_3
O
1
O
2
O
3
. Prove that
r
=
r
1
r
2
r
3
r
1
+
r
2
+
r
3
.
r=\sqrt{\dfrac{r_1r_2r_3}{r_1+r_2+r_3}}.
r
=
r
1
+
r
2
+
r
3
r
1
r
2
r
3
.
geometry
tangent circles