document.write( "Question 1148562: In the triangle below, one vertex is joined to two points on the opposite side of the triangle, another is joined to four points on the opposite side, and another is joined to six points on the opposite side. No three lines intersect at one point. How many non-overlapping regions are formed in the triangle. \r
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Algebra.Com's Answer #769946 by ikleyn(52781) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "\n" ); document.write( "I do not think that the original formulation is correct.
\n" ); document.write( "In this formulation, it is not a Math problem, at all. It is the problem for a \"kindergarten\", conditionally speaking.
\n" ); document.write( "To be a Math problem, it should ask not about non-overlapping regions, but about ALL CONVEX regions in the triangle.\r
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\n" ); document.write( "\n" ); document.write( "            Reformulated in this way, the problem is of the school/local university Math circle level or local Math Olympiad level.\r
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\n" ); document.write( "\n" ); document.write( "            I will not go into details --- instead, I will give you an idea of the solution only,
\n" ); document.write( "            assuming that your level is adequate and you will be able to complete the solution on your own.\r
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\n" ); document.write( "\n" ); document.write( "            Before to start, let me note, that, in accordance with the context, the problem should ask about the CONVEX \r
\n" ); document.write( "\n" ); document.write( "            regions (polygons) ONLY.\r
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\n" ); document.write( "\n" ); document.write( "            We are given 2 points on one side of the triangle; with its vertices, this side has 4 points, and I will call it 4-point side.\r
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\n" ); document.write( "\n" ); document.write( "            Similarly, we have 6-point side and 8-point side.\r
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document.write( "The regions the problem asks for, can be a) triangles; b) quadrilaterals; c) pentagons; and d) hexagons ONLY.\r\n" );
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document.write( "They can not have more than 6 sides.\r\n" );
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document.write( "Each triangle is formed as the intersection of 3 lines: \r\n" );
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document.write( "    - one line to 4-point side;\r\n" );
document.write( "    - one line to 6-point side, and\r\n" );
document.write( "    - one line to 8-point side.\r\n" );
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document.write( "More precisely, there is ONE-to-ONE correspondence between choosing triples of such lines and triangles.\r\n" );
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document.write( "So, the number of triangles is   =  = 64.\r\n" );
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document.write( "Each quadrilateral is formed as the intersection of 4 lines\r\n" );
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document.write( "    - two lines on 4-point side; one line to 6-point side and one line to 8-point side; in all, there are  such quadrilaterals;\r\n" );
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document.write( "    - one line on 4-point side; two lines to 6-point side and one line to 8-point side; in all, there are  such quadrilaterals;\r\n" );
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document.write( "    - one line on 4-point side; one line to 6-point side and two lines to 8-point side; in all, there are  such quadrilaterals.\r\n" );
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document.write( "Having this, you can complete calculating the number of quadrilaterals.\r\n" );
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document.write( "    (When you complete calculating the number of combination of lines, do not forget to divide this number by 4! = 24.\r\n" );
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document.write( "      It is because we consider non-ordered combinations of lines, ONLY (!) )\r\n" );
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document.write( "The idea of solution and the pattern should be clear to you now.\r\n" );
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document.write( "Next you should analyse pentagons and hexagons by the similar way.\r\n" );
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\n" ); document.write( "\n" ); document.write( "At this point I complete my instructions.\r
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\n" ); document.write( "\n" ); document.write( " The lesson to learn from my instructions and from the solution is THIS:\r
\n" ); document.write( "\n" ); document.write( " - there is ONE-to-ONE correspondence between these convex regions/polygons and combinations of lines.\r
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\n" ); document.write( "\n" ); document.write( "I wish good luck to you with the further solution.\r
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\n" ); document.write( "\n" ); document.write( "If you have difficulties or question in this way, then let me know,
\n" ); document.write( "referring to the problem's ID number 1148562.\r
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