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2

Part of 1995 Balkan MO

Problems(1)

Again geometry involving two circles that intersect

Source: Balkan MO 1995, Problem 2

4/24/2006
The circles C1(O1,r1)\mathcal C_1(O_1, r_1) and C2(O2,r2)\mathcal C_2(O_2, r_2), r2>r1r_2 > r_1, intersect at AA and BB such that ∠O1AO2=90∘\angle O_1AO_2 = 90^\circ. The line O1O2O_1O_2 meets C1\mathcal C_1 at CC and DD, and C2\mathcal C_2 at EE and FF (in the order CC, EE, DD, FF). The line BEBE meets C1\mathcal C_1 at KK and ACAC at MM, and the line BDBD meets C2\mathcal C_2 at LL and AFAF at NN. Prove that r2r1=KEKM⋅LNLD. \frac{ r_2}{r_1} = \frac{KE}{KM} \cdot \frac{LN}{LD} . Greece
geometryBMO