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Synthetic Geometry Group-Nihar's Proposal

These problems are my submissions to Xuming's Geometry group.These problems are taken from previous RMO papers so that some of my friends who are preparing for RMO will be benefited by this discussion and thereby prepare and improve themselves :)

Q1) Let \(AL\) and \(BK\) be the angle bisectors in a non-isosceles triangle \(ABC,\) where \(L\) lies on \(BC\) and \(K\) lies on \(AC.\) The perpendicular bisector of \(BK\) intersects the line \(AL\) at \(M\). Point \(N\) lies on the line \(BK\) such that \(LN\) is parallel to \(MK.\) Prove that \(LN=NA.\)

Q2) Let \(ABC\) be a triangle and let \(BB_1,CC_1\) be respectively the bisectors of \(\angle{B},\angle{C}\) with \(B_1\) on \(AC\) and \(C_1\) on \(AB\), Let \(E,F\) be the feet of perpendiculars drawn from \(A\) onto \(BB_1,CC_1\) respectively. Suppose \(D\) is the point at which the incircle of \(ABC\) touches \(AB\). Prove that \(AD=EF\)

Note by Nihar Mahajan
1 year, 4 months ago

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Since we know that in a triangle the perpendicular bisector of a side and angular bisector of the angle opposite to this side meet at a point which is concyclic with the vertices of the triangle. Here, angular bisector of \(\angle BAK\) and perpendicular bisector of \( BK\) meet at \(M\). So, \(A\), \(B\), \(M\) and \(K\) are concyclic. So, \(\angle AMK = \angle ABK\).

But, \(LN || MK\). So, \(\angle ALN = \angle AMK = \angle ABK = \angle ABN\). So, \(A\), \(B\), \(L\) and \(N\) are cyclic. It implies that \(\angle NAL = \angle NBL = \angle NBA = \angle ALN\). So, \(\angle ALN = \angle ANL\), which implies that \(NA = NL\). Surya Prakash · 1 year, 4 months ago

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@Surya Prakash Yeah , this is the standard solution. At first the problem looks tough. But as you decipher the configuration , you crack it like a left hand's play (provided that you are right handed :P) Nihar Mahajan · 1 year, 4 months ago

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@Surya Prakash what rank did you get in jee 2016 ? Shubham Dhull · 4 months, 4 weeks ago

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@Surya Prakash @Mehul Arora @Agnishom Chattopadhyay @Alan Yan @Shivam Jadhav @Ambuj Shrivastava @Swapnil Das @Sharky Kesa @Saarthak Marathe @Kushagra Sahni @naitik sanghavi If you are interested in this , please make your proposals soon and await the geometry challenges,discussion!

@Calvin Lin Sorry , for mass tagging. But I think this will encourage the participation :) Nihar Mahajan · 1 year, 4 months ago

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@Nihar Mahajan I made mine Shivam Jadhav · 1 year, 3 months ago

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@Nihar Mahajan :3 :3 :3 :3 :3 :3 :3 :3 :3 Mehul Arora · 1 year, 4 months ago

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modern algebra text Gebretsadkan Gebereyohannes · 1 year, 2 months ago

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Here's mine @Nihar Mahajan - Surya Prakash's Proposals Surya Prakash · 1 year, 4 months ago

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@Calvin Lin @Xuming Liang Here's my submission! I hope that this marvelous group forms and accelerates soon :) Nihar Mahajan · 1 year, 4 months ago

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