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 $SP=SC$.