MathDB
How many points is too many

Source: RMM 2021/1

October 13, 2021
geometryRMM

Problem Statement

Let T1,T2,T3,T4T_1, T_2, T_3, T_4 be pairwise distinct collinear points such that T2T_2 lies between T1T_1 and T3T_3, and T3T_3 lies between T2T_2 and T4T_4. Let ω1\omega_1 be a circle through T1T_1 and T4T_4; let ω2\omega_2 be the circle through T2T_2 and internally tangent to ω1\omega_1 at T1T_1; let ω3\omega_3 be the circle through T3T_3 and externally tangent to ω2\omega_2 at T2T_2; and let ω4\omega_4 be the circle through T4T_4 and externally tangent to ω3\omega_3 at T3T_3. A line crosses ω1\omega_1 at PP and WW, ω2\omega_2 at QQ and RR, ω3\omega_3 at SS and TT, and ω4\omega_4 at UU and VV, the order of these points along the line being P,Q,R,S,T,U,V,WP,Q,R,S,T,U,V,W. Prove that PQ+TU=RS+VWPQ + TU = RS + VW
Geza Kos, Hungary
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