document.write( "Question 1187405: https://imgur.com/HQvOuKW In the triangle, each vertex is joined to four points on the opposite side of the triangle, with no three lines intersecting at one point. How many non-overlapping regions are formed in the triangle? \n" ); document.write( "

Algebra.Com's Answer #818518 by ikleyn(52780) $\"\"$ $\"About$

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\n" ); document.write( "In the triangle, each vertex is joined to four points on the opposite side of the triangle,
\n" ); document.write( "with no three lines intersecting at one point.
\n" ); document.write( "How many non-overlapping regions are formed in the triangle?
\n" ); document.write( "https://imgur.com/HQvOuKW
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\n" ); document.write( "\n" ); document.write( "            It may seem unexpected, but this problem has a nice mathematical solution.\r
\n" ); document.write( "\n" ); document.write( "            It uses one of the most beautiful formulas of Mathematics.\r
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\n" ); document.write( "\n" ); document.write( "            The greatest mathematician Euler will come to us to help solving this problem.\r
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\n" ); document.write( "\n" ); document.write( "            I will use the Euler formula for polygonal grids in the plane.\r
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\n" ); document.write( "\n" ); document.write( "            Applause and cheers to Great Euler ( ! )\r
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document.write( "The Euler formula, connecting the number of faces F, edges E and vertices V of a convex polyhedron \r\n" );
document.write( "in 3D is widely known.  It is  F - E + V = 2.\r\n" );
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document.write( "In our case, we have a polygonal grid on a plane; in this case, the Euler formula is \r\n" );
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document.write( "    F - S + V = 1,     (1)\r\n" );
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document.write( "where F is the number of faces (minimal polygons of the grid); S is the number of their sides and\r\n" );
document.write( "V is the number of vertices.\r\n" );
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document.write( "We will calculate the numbers S and V in the formula (1), and then use it to determine F, our major unknown.\r\n" );
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document.write( "Part 1.  Calculating the number of vertices V\r\n" );
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document.write( "    Let call our triangle ABC, by the names of its vertices.\r\n" );
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document.write( "    We have 4 lines from vertex A to the opposite side \"a\".\r\n" );
document.write( "    We have 4 lines from vertex B to the opposite side \"b\".\r\n" );
document.write( "    These two families of lines have 4 x 4 = 16 intersection points.\r\n" );
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document.write( "    Next, we have 4 lines from vertex A to the opposite side \"a\",\r\n" );
document.write( "     and  we have 4 lines from vertex C to the opposite side \"c\".\r\n" );
document.write( "    These two families of lines produce other 4 x 4 = 16 intersection points, different from 16 points above.\r\n" );
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document.write( "    \r\n" );
document.write( "    Finally, we have 4 lines from vertex B to the opposite side \"b\",\r\n" );
document.write( "        and  we have 4 lines from vertex C to the opposite side \"c\".\r\n" );
document.write( "    These two families of lines produce other 4 x 4 = 16 intersection points, different from 32 points above.\r\n" );
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document.write( "    So, we have 16 + 16 + 16 = 48 intersection points INSIDE the triangle ABC.\r\n" );
document.write( "    To it, we should add 5 + 5 + 5 = 15 intersection points along the PERIMETER of the triangle ABC.\r\n" );
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document.write( "    In all, there are  48 + 15 = 63 intersection points:  V = 63.\r\n" );
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document.write( "Part 2.  Calculating the number of sides S\r\n" );
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document.write( "    \r\n" );
document.write( "    From vertex A to the opposite side \"a\" we have 4 interior lines inside triangle ABC.\r\n" );
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document.write( "    Interesting fact is that each of these lines has 9 elementary interior segments.\r\n" );
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document.write( "        You may count and check this fact on your own.\r\n" );
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document.write( "    It is not an accidental fact: 9 elementary interior segments in each of these lines are created\r\n" );
document.write( "    by 4 + 4 = 8 intersection points of this line with the family of 4 lines  \" from B to b \"  and\r\n" );
document.write( "    with the family of 4 lines  \" from C to c \".\r\n" );
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document.write( "    So, the four lines  \" from A to a \"  give us 4*9 = 36 elementary segments.\r\n" );
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document.write( "    Similarly, from vertex B to the opposite side \"b\" we have 4 interior lines inside triangle ABC.\r\n" );
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document.write( "    Again, interesting fact is that each of these lines has 9 elementary interior segments.\r\n" );
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document.write( "        You may count and check this fact on your own.\r\n" );
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document.write( "    The reason is similar to the above case.\r\n" );
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document.write( "    So, the four lines  \" from B to b \"  give us 4*9 = 36 another elementary segments.\r\n" );
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document.write( "    Finally and similarly, we have another 36 elementary segments from the family of lines \" from C to c \".\r\n" );
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document.write( "    In all, we just counted  36 + 36 + 36 = 108 elementary segments INSIDE the triangle ABC.\r\n" );
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document.write( "    Add to it 5 + 5 + 5 = 15 elementary segments along the PERIMETER of triangle ABC.\r\n" );
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document.write( "    So, the total number of elementary segments is 108 + 15 = 123:  S = 123.\r\n" );
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document.write( "Part 3.  Applying the Euler formula\r\n" );
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document.write( "    From the formula (1),  we have\r\n" );
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document.write( "        F = 1 + S - V = 1 + 123 - 63 = 61.\r\n" );
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document.write( "ANSWER.  The number of faces inside the triangle ABC (non-overlapping regions) is 61.\r\n" );
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