MathDB

3

Part of 2002 Moldova Team Selection Test

Problems(3)

Three circles externally tangent - M, N, and L are collinear

Source: Moldova TST 2002 - E1 - P3

8/19/2009
The circles W1,W2,W3W_1, W_2, W_3 in the plane are pairwise externally tangent to each other. Let P1P_1 be the point of tangency between circles W1W_1 and W3W_3, and let P2P_2 be the point of tangency between circles W2W_2 and W3W_3. AA and BB, both different from P1P_1 and P2P_2, are points on W3W_3 such that ABAB is a diameter of W3W_3. Line AP1AP_1 intersects W1W_1 again at XX, line BP2BP_2 intersects W2W_2 again at YY, and lines AP2AP_2 and BP1BP_1 intersect at ZZ. Prove that X,YX, Y, and ZZ are collinear.
geometric transformationgeometryhomothetycyclic quadrilateralgeometry proposed
midpoints of 2 arcs, point on arc,2 incenters, similar triangles wanted

Source: Moldova 2002 TST 2 P3

8/25/2018
A triangle ABCABC is inscribed in a circle GG. Points MM and NN are the midpoints of the arcs BCBC and ACAC respectively, and DD is an arbitrary point on the arc ABAB (not containing CC). Points I1I_1 and I2I_2 are the incenters of the triangles ADCADC and BDCBDC, respectively. If the circumcircle of triangle DI1I2DI_1I_2 meets GG again at PP, prove that triangles PNI1PNI_1 and PMI2PMI_2 are similar.
geometryincentersimilar trianglesarc
locus of points M so BM x CM / MA_1 is minimal, A_1 = AM \cap (ABC)

Source: Moldova 2002 TST 3 P3

8/25/2018
A triangle ABCABC is inscribed in a circle GG. For any point MM inside the triangle, A1A_1 denotes the intersection of the ray AMAM with GG. Find the locus of point MM for which BM⋅CMMA1\frac{BM\cdot CM}{MA_1} is minimal, and find this minimum value.
geometryLocusminimum