December 5th, 2017, 11:09 PM  #1 
Newbie Joined: Dec 2017 From: Iran Posts: 3 Thanks: 0  geometry proof triangle
It's an equilateral triangle named "ABC" The center point of the triangle is named "O" The circle that crosses "A" point and "O" crosses "AB" in "M" point and "AC" in "N" point . Prove that AN = BM. Last edited by skipjack; December 6th, 2017 at 06:43 PM. 
December 6th, 2017, 02:24 PM  #2 
Global Moderator Joined: May 2007 Posts: 6,398 Thanks: 546 
Equilateral triangle is congruent to itself by flipping along OA line.

December 6th, 2017, 07:56 PM  #3 
Global Moderator Joined: Dec 2006 Posts: 18,248 Thanks: 1439 
OA = OB (given) ON = OM (because $\small\angle$NAO = $\small\angle$MAO = 30$^\circ$) $\small\angle$AON = $\small\angle$BOM (because $\small\angle$MON = $\small\angle$BOA = 120$^\circ$) Hence triangles ANO, BMO are congruent (SAS), which implies AN = BM. 

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