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.
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.
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).
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 \(AP= AQ\).
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 \(SP=SC\).