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Part of 2013 North Korea Team Selection Test

Problems(1)

Concurrent lines

Source: North Korea Team Selection Test 2013 #1

5/17/2014
The incircle of a non-isosceles triangle ABCABC with the center II touches the sides BC,CA,AB BC, CA, AB at A1,B1,C1 A_1 , B_1 , C_1 respectively. The line AIAI meets the circumcircle of ABCABC at A2A_2 . The line B1C1B_1 C_1 meets the line BCBC at A3A_3 and the line A2A3A_2 A_3 meets the circumcircle of ABCABC at A4(≠A2)A_4 (\ne A_2 ) . Define B4,C4B_4 , C_4 similarly. Prove that the lines AA4,BB4,CC4 AA_4 , BB_4 , CC_4 are concurrent.
geometrycircumcircletrigonometryNorth Korea