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Synthetic Geometry Group - Surya Prakash's Proposal

This is my submission to Xuming's Synthetic Geometry Group. These problems are taken from different Olympiads. Try them on your own. I will soon post the solutions. Feel free to post the solutions.

  1. Let \(\Gamma\) be the circumcircle of \(\Delta ABC\), and let \(l\) be the tangent of \(\Gamma\) passing through \(A\). Let \(D\), \(E\) be the points on side \(AB\) and \(AC\) such that \(BD :DA = AE : EC\). Line \(DE\) meets \(\Gamma\) at points \(F\), \(G\). The line parallel to \(AC\) passing \(D\) meets \(l\) at \(H\), the line parallel to \(AB\) passing \(E\) meets \(l\) at \(I\). Prove that \(F\), \(G\), \(H\), \(I\) are cyclic and BC is tangent to the circle through these points.

  2. In \(\Delta ABC\), let \(H\) be the orthocenter of the triangle and \(M\) be the midpoint of the side \(BC\). Let the line perpendicular to \(HM\) through \(H\) meet \(AC\) and \(AB\) at \(E\) and \(F\). Prove that \(HE = HF\). (Proposed by Xuming).

  3. Let \(\Delta ABC\) be an acute triangle with \(D\), \(E\), \(F\) the feet of the altitudes lying on \(BC\), \(CA\), \(AB\) respectively. One of the intersection points of the line \(EF\) and the circumcircle is \(P\). The lines \(BP\) and \(DF\) meet at point \(Q\). Prove that \(AP= AQ\).

  4. Let \(P\) be a point inside triangle \(\Delta ABC\). Lines \(AP\), \(BP\), \(CP\) meet the circumcircle of \(\Delta ABC\) again at points \(K\), \(L\), \(M\) respectively. The tangent to the circumcircle at \(C\) meets line \(AB\) at \(S\). Prove that \(MK=ML\) if and only if \(SP=SC\).

Note by Surya Prakash
2 years ago

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@Nihar Mahajan @Mehul Arora @Aditya Kumar @Xuming Liang @Calvin Lin @Shivam Jadhav @Swapnil Das @Adarsh Kumar @Sharky Kesa

Please respond to this proposals. and post your proposals too.

Surya Prakash - 2 years ago

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Nice ones man!

Adarsh Kumar - 2 years ago

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Nice problems!

Aditya Kumar - 2 years ago

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This proposal*

Will check it out :)

Mehul Arora - 2 years ago

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is the condition if and only if in question 4 right?

Onkar Tiwari - 1 year, 9 months ago

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Nicce problems! I just want to point out that I did not propose that problem. It can be viewed as a simple application of the Butterfly theorem(do you see it?). A "pesudo" generalization of this was utilized in one of my recent problems though.

Xuming Liang - 2 years ago

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no problem. by the way I got the solution for that.

Surya Prakash - 2 years ago

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