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0|0|0|4c0e90da3a7466af.jpg||||Another Solution For Cutting Triangles off Regular Hexagon|Bui Quang Tuan||14:17:42|06/08/2010|Dear Alex%2C%0D%0A%0D%0AMay be following my solution for problem %22Cutting Triangles off Regular Hexagon%22%3A%0D%0A%0D%0Ahttp%3A%2F%2Fwww.cut-the-knot.org%2Ftriangle%2FMidpointsInHexagon.shtml%0D%0A%0D%0Ais interesting.%0D%0A%0D%0A%28Please see my attached image%29%0D%0ASuppose%2C ABCDEF is regular hexagon and M%2C N%2C P%2C Q%2C R%2C S are midpoints of AB%2C BC%2C CD%2C DE%2C EF%2C FA%0D%0AOf course%2C by similarities%2C Area%28ABCDEF%29%2FArea%28MNPQRS%29 %3D Area%28BCEF%29%2FArea%28MPQS%29%0D%0AIn the triangle ABF%3A %0D%0AMS %3D BF%2F2%0D%0AIn the trapezium ABCD%3A%0D%0AMP %3D 3%2ABC%2F2%0D%0ATherefore%3A%0D%0AArea%28ABCDEF%29%2FArea%28MNPQRS%29 %3D Area%28BCEF%29%2FArea%28MPQS%29 %3D BF%2ABC%2F%28MS%2AMP%29 %3D 4%2F3%0D%0A%0D%0ABest regards%2C%0D%0ABui Quang Tuan
1|1|0|||||RE%3A Cutting Triangles off Regular Hexagon|alexb||15:05:11|06/08/2010|Thank you.%0D%0A%0D%0ATime to relax. I am going to add another one. %0D%0A%0D%0A%0D%0A%0D%0A
2|1|0|4c0fa79d390e3801.jpg||||RE%3A Another Solution For Cutting Triangles...|Bui Quang Tuan||11:39:47|06/09/2010|Dear Alex%2C%0D%0A%0D%0APlease see my another attached image%0D%0A%0D%0AWe can avoid using similarities because %0D%0AArea%28ABCDEF%29%2FArea%28BCEF%29 %3D Area%28MNPQRS%29%2FArea%28MPQS%29 %3D 3%2F2%0D%0A%0D%0AWe can also have another solution by construction of points IJKH.%0D%0A%0D%0AArea%28MNPQRS%29%2FArea%28MPQS%29 %3D 3%2F2%0D%0A%0D%0AArea%28ABCDEF%29%3DArea%28IJKH%29%0D%0AArea%28ABCDEF%29%2FArea%28MPQS%29 %3D Area%28IJKH%29%2FArea%28BCEF%29 %3D 2%0D%0A%0D%0AArea%28ABCDEF%29%2FArea%28MNPQRS%29 %3D 2%2F%283%2F2%29 %3D 4%2F3%0D%0A%0D%0ABest regards%2C%0D%0ABui Quang Tuan%0D%0A
3|2|2|||||RE%3A Another Solution For Cutting Triangles...|alexb||12:18:11|06/09/2010|Just great. It%27s all a trifle%2C but I am enjoying myself. Thank you very much for lending a hand.%0D%0A%0D%0AAlex
4|1|0|4c17b25a10269dc8.jpg||||RE%3A Another Solution For Cutting Triangles...|Bui Quang Tuan||12:07:19|06/15/2010|Dear Alex%2C%0D%0A%0D%0AThe area ratio 4%2F3 is still true when ABCDEF is one convex hexagon with symmetric center%2C say O. Please see attached image%21%0D%0A%0D%0AIn the triangle AEC%0D%0AArea%28OCE%29%3DArea%28OEF%29%0D%0AArea%28OAC%29%3DArea%28OCD%29%0D%0AArea%28OAE%29%3DArea%28OED%29%0D%0ATherefore Area%28ACE%29%3DArea%28FEDC%29%3DArea%28ABCDEF%29%2F2%0D%0AArea%28ABCDEF%29%3D2%2AArea%28ACE%29 %281%29%0D%0A%0D%0AIn the trapezium SMNR%0D%0ASM%3DFB%2F2%3DEC%2F2%0D%0ARN%3DEC%0D%0A%0D%0ADistance from SM to FB %3D 1%2F2 Distance from A to FB%0D%0ADistance from FB to RN %3D 1%2F2 Distance from FB to EC%0D%0ATherefore%0D%0ADistance from SM to RN %3D 1%2F2 Distance from A to EC%0D%0Aand%0D%0AArea%28SMNR%29%3D3%2F4%2AArea%28ACE%29%0D%0AWe also have Area%28SMNR%29%3DArea%28MNPQRS%29%2F2 so%0D%0AArea%28MNPQRS%29%3D2%2AArea%28SMNR%29%3D3%2F2%2AArea%28ACE%29 %282%29%0D%0A%0D%0AFrom %281%29%2C %282%29 we have result%3A%0D%0AArea%28ABCDEF%29%2FArea%28MNPQRS%29%3D4%2F3%0D%0A%0D%0ABest regards%2C%0D%0ABui Quang Tuan
6|2|4|4c19219f18e4c39b.jpg||||RE%3A Another Solution For Cutting Triangles...|Bui Quang Tuan||14:17:00|06/16/2010|Dear Alex%2C%0D%0A%0D%0AWe can have more nice proof for this fact as for regular hexagon case. Please see attached image%21%0D%0A%0D%0AIn the triangle OMN%3A%0D%0ASuppose K is midpoint of AC. Easy to show three following congruent triangles%3A%0D%0AKNO %3D AMS%0D%0AKMO %3D CNP%0D%0AKMN %3D BNM%0D%0ASimilarly we can do with other triangles ONP%2C OPQ%2C OQR%2C ORS%2C OSM and we have attached image. All triangles with same color are congruent.%0D%0AFrom this image%3A%0D%0AArea%28ABCDEF%29%2FArea%28MNPQRS%29%3D4%2F3%0D%0A%0D%0ABest regards%2C%0D%0ABui Quang Tuan
7|3|6|||||RE%3A Another Solution For Cutting Triangles...|alexb||14:45:16|06/16/2010|As beautiful as they come. Thank you.