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As is well-known, a Cevian triangle is the triangle formed by the intersections of a triangle's sides with their corresponding Cevians (a line through a point inside the triangle, i.e. the Cevian point, and the vertex opposite the side).

An application I'm considering requires me to reckon the Cevian triangle of smallest area with respect to the centroid of the triangle that is similar to the original triangle. For the equilateral case, the medial triangle is an "obvious" solution, but is the medial triangle the Cevian triangle with respect to the centroid that is similar and has the smallest area, or are there more "optimal" triangles?

I've tried searching around, but I am probably not using the right keywords. Any help would be lovely.

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    If you know the Cevian point, I would have though that there was only one Cevian triangle, and for the centroid it is the medial triangle (which, unlike most Cevian triangles, is similar to the original triangle). So what is the question?2011-04-11
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    You can have Cevian triangles which are similar to the original triangle but smaller than the medial triangle, though not related to the centroid. Take the triangle $(0,0)$, $(1,0)$, $(0,1)$: the centroid is about $(0.333,0.333)$ and the medial triangle $(0.5,0)$, $(0,0.5)$, $(0.5,0.5)$ has area $0.125$. However with a Cevian point of about $(0.179509025, 0.30437923)$, the Cevian triangle $(0.258055872,0)$, $(0,0.370972064)$, $(0.370972064,0.629027936)$ [all roots of cubic equations] is similar but has area $0.102106553$. I doubt there are more than 6 similar Cevian triangles.2011-04-11

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