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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
1 year, 1 month ago

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Please respond to this proposals. and post your proposals too. · 1 year, 1 month ago

Nice ones man! · 1 year, 1 month ago

Nice problems! · 1 year, 1 month ago

This proposal*

Will check it out :) · 1 year, 1 month ago

is the condition if and only if in question 4 right? · 9 months, 4 weeks ago

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. · 1 year, 1 month ago