Dear Mathlinkers, 1. A, B the end points of an arch circle 2. (O) a circle tangent to AB intersecting the arch in question 3. T the point of contact of (O) and AB 4. C, D the points of intersection of (O) with the arch in the order A, D, C, B 5. E, F the points of intersection of AC and DT, BD and CT. Prove : EF is parallel to AB. Sincerely Jean-Louis
Problem
Source: Sharygin olympiad 2013
Tags: geometry proposed, geometry
jayme
15.02.2014 17:03
Dear Mathlinkers no ideas for this 8 grade problem? Sincerely Jean-Louis
fmasroor
15.02.2014 18:50
Can you give a diagram please?
jayme
16.02.2014 09:29
Dear Mathlinkers, 1. extend CT and DT until the complete circle including the arch AB 2. C', D' are the second points of intersection 3. according to Teim's theorem, C'D' // AB the rest follows... Sincerely Jean-Louis
fmasroor
16.02.2014 18:26
My quick proof using infiniteturtle's diagram... <TDB+<BDC=<TDC=<BTC=<BAC+<TCA=<BDC+<TCA so <TDB=<TCA so DCEF is cyclic, then <BTC=<TDC=<EFT and so EF || AB