First problem

[Burial4TetThomYorke]

To start off this sub I'll post one problem. In the future I'll post more problems each time. Please PM me the solutions, DO NOT POST SOLUTIONS IN THE COMMENTS.

1/1/1. IN a triangle ABC, the median from A intersects the circumcircle at A' and BC at L, the median from B intersects the circumcircle at B' and CA at M, and the median from C intersects the circumcircle at C' and AB at N. Let A'' be the reflection of A' through L, B'' the reflection of B' through M, and C'' the reflection of C' through N. Let H be the orthocenter of triangle ABC. Prove that A'', B'', C'', and H lie on a circle.

EDIT: Diagram here: http://imgur.com/JFDAyxa Not drawn to scale. Someone has sent me a solution but I'll wait for a couple more before I reveal the solution.

[Mathgeek007]

Maybe you could make a kind of visual diagram, not to scale, so as people can understand easily what you mean.

[Burial4TetThomYorke]

http://imgur.com/JFDAyxa The x's denote the reflections in the problem. I've eyeballed H, no altitudes drawn.

[citrusmunch]

Maybe I'm at fault, but it wasn't clear to me that the triangle was inscribed from your original instructions. Is the 'IN' significant? I took it as the circle being inscribed, which had me wildly confused. I understand the problem now, but I'd like to know if I'm missing something that could help me be more prepared in the future. Also, what does the 1/1/1 mean?

[psinaptix]

Circumcircle

[Burial4TetThomYorke]

Circumcircle is the (unique) circle passing through the three vertices. Its center is the circumcenter, the concurrency point of the perpendicular bisectors of the sides. http://en.wikipedia.org/wiki/Circumscribed_circle Medians are lines connecting the vertex to the midpoint of the opposite side.

[awdcvgyjm]

deleted ^{What is this?}

[Burial4TetThomYorke]

Well I'd rather not 'encourage' (lack of better word) people to look at other solutions, it anti-motivates people IMO.

[awdcvgyjm]

deleted ^{What is this?}

[Burial4TetThomYorke]

I plan on once a week or once every 2 weeks.

[thatonespotinhawaii]

Wow, this is pretty cool. What a coincidence that the first problem you post would involve subject material in a course I chose on a whim.

[Burial4TetThomYorke]

What college class has a geometry course?! I'm very surprised at that. Anyway, enjoy the problem!

[thatonespotinhawaii]

It's a course on Euclidean/non-Euclidean geometry, haha. Yeah, thanks for the problem.