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312

Part of 1955 Moscow Mathematical Olympiad

Problems(1)

MMO 312 Moscow MO 1955 equal rations lead to parallels, Δ inscribed in Δ

Source:

8/17/2019
Given △ABC\vartriangle ABC, points C1,A1,B1C_1, A_1, B_1 on sides AB,BC,CAAB, BC, CA, respectively, such that AC1C1B=BA1A1C=CB1B1A=1n\frac{AC_1}{C_1B}= \frac{BA_1}{A_1C}= \frac{CB_1}{B_1A}=\frac{1}{n} and points C2,A2,B2C_2, A_2, B_2 on sides A1B1,B1C1,C1A1A_1B_1, B_1C_1, C_1A_1 of △A1B1C1\vartriangle A_1B_1C_1, respectively, such that A1C2C2B1=B1A2A2C1=C1B2B2A1=n\frac{A_1C_2}{C_2B_1}= \frac{B_1A_2}{A_2C_1}= \frac{C_1B_2}{B_2A_1}= n. Prove that A2C2//AC,C2B2//CB,B2A2//BAA_2C_2 //AC, C_2B_2 // CB, B_2A_2 // BA.
geometryequal ratiosparallel