$geometry olympiad problems$

GEOMETRY OLYMPIAD PROBLEMS SOFTWARE

In this narrative, the point names might be a little strange at first, because (to make the story follow-able) I used the point names that ended up in the final problem, rather than ones I initially gave. None of this makes sense written this abstractly, so now let me walk you through the two problems I wrote. Or shorter yet: build up, then tear down. Look for other ways to reduce the number of points even further, by finding other equivalent formulations that have fewer points.Pick the one that is most elegant (or hardest), and erase auxiliary points you added that are not needed for the problem statement. Once you are happy with what you have, you have a nontrivial statement and probably several things that are equivalent to it.Repeat the previous two steps to your satisfaction.This will probably add more points to you figure, since you often need to construct more auxiliary points to prove the conjecture that you have found. Figure out why this coincidence happened.Perturb the diagram to make sure your conjecture looks like itвЂ™s true in all cases. Keep going with this until you find something unexpected: three points are collinear, four points are cyclic, or so on.

You might be guided by some actual geometry constructions: for example, if you know that the starting configuration has a harmonic bundle in it, you might project this bundle to obtain the new points to play with.

Start playing around, adding in points and so on to see if anything interesting happens.

For example, you might draw a triangle with its incircle and then add in the excircle tangency point, and the circle centered at passing through both points (taking advantage of the fact that the two tangency points are equidistant from and ).

Start with a configuration of your choice something that has a bit of nontrivial structure in it, and add something more to it.

With that in mind, hereвЂ™s the gist of what you do.

GEOMETRY OLYMPIAD PROBLEMS SOFTWARE

Geogebra, or some other software that will let you quickly draw and edit diagrams.Fortunately, this is something youвЂ™ll hopefully have just from having done enough olympiad problems. In particular, a good eye: in an accurate diagram, you should be able to notice if three points look collinear or if four points are concyclic, and so on.The intuition you have from being a contestant proves valuable when you go about looking for things. The ability to consistently solve medium to hard olympiad geometry problems.Here are the main ingredients youвЂ™ll need. Prove that the circle with diameter is orthogonal to the nine-point circle of. The tangents to the circumcircle of at and meet at. In scalene triangle with incenter, the incircle is tangent to sides and at points and. Let be a point on the incircle such that. Denote by the midpoint of and let be a point in the interior of so that and. Let be a triangle with incenter whose incircle is tangent to, , at, ,, respectively. Note that I donвЂ™t claim this is the only way to write such problems, it just happens to be the approach I use, and has consistently gotten me reasonably good results. In particular, IвЂ™ll go into detail about how I created the following two problems, which were the first olympiad problems which I got onto a contest. In this post IвЂ™ll describe how I come up with geometry proposals for olympiad-style contests. The combination is also up to you so choose the ones you like most. Consider nettles, beet tops, turnip tops, spinach, or watercress in place of chard. You can use a wide range of wild, cultivated or supermarket greens in this recipe.

Geometry olympiad problems