Peter (pleurestique) wrote in moppers,
Peter
pleurestique
moppers

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two triangles and a point

though i never made MOP (i was close, though, to being a USAMO winner my senior year), i just joined this community today because i'm once again interested in math (after turning away from it for about a year -- my attitude towards it changes periodically), *and* i wanted to post this problem to y'all. oh, and ayo lawrence!


Alfred S. Posamentier gave me this problem (said it was given to him by a guy named Hauptman; i forgot the guy's first name; though the name Hauptman in connection to geometry sounds damn familiar to me):

Given two triangles and a point -- all in the same plane -- prove that there exists a triangle inscribed in one of the triangles that's similar to the other triangle and that has a side (or its extension) passing through the given point.

now, Posmo (as we so affectionately call him) *might* have left out a few details and stipulations, as he stated the problem to me oh so casually and colloquially... it seems like the assertion is false for certain configurations... for example, if the two triangles overlap "a lot" and if the point is located in the intersection of the two sets: 1) the intersection of the interiors of all triangles inscribed in the first triangle and similar to the second; 2) the intersection of the interiors of all triangles inscribed in the second and similar to the first. (it seems easy to have these two sets intersect non-emptily, given sufficiently overlapping triangles to begin with)

so, now, let me ask this, people... aside from this uptight kind of configuration... is the assertion possibly true for "most" configurations? what must we stipulate to have it be true? and hopefully it's an easily stated stipulation... no more than an *eighth* of the way to stipulating the *very* circumstance we desire! (whatever "an eighth of the way" means... though talking in this way is obviously done ironically connecting the euclidean plane of speaking and the hyperbolic one of speaking... *obviously*)

no, and i really *am* serious about this question... i hope someone will have something to say...
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