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?
Found 2 solutions by greenestamps, ikleyn:
Answer by greenestamps(13200)   (Show Source):
You can put this solution on YOUR website!
There might be some fancy mathematical formula for finding the answer to a problem like this, but it's probably more complicated than finding the answer by just counting.
What is required is a systematic method for counting the regions so that each region gets counted exactly once. Here is what I suggest.
(1) Pick one vertex of the big triangle and consider the 6 lines from that vertex to the opposite side (2 lines to the other two vertices, and 4 lines to the opposite side). Those lines divide the big triangle into 5 triangular regions.
(2) Count (carefully, and perhaps at least twice) the number of regions in each of those 5 triangular regions. As a check on your counting, note that, by symmetry, the numbers of regions in the 1st and 5th triangular regions should be the same, as should the numbers of regions in the 2nd and 4th.
Happy counting!
NOTE: My count was 61
Answer by ikleyn(52779)  (Show Source):
You can put this solution on YOUR website! .
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?
https://imgur.com/HQvOuKW
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It may seem unexpected, but this problem has a nice mathematical solution.
It uses one of the most beautiful formulas of Mathematics.
The greatest mathematician Euler will come to us to help solving this problem.
I will use the Euler formula for polygonal grids in the plane.
Applause and cheers to Great Euler ( ! )
The Euler formula, connecting the number of faces F, edges E and vertices V of a convex polyhedron
in 3D is widely known. It is F - E + V = 2.
In our case, we have a polygonal grid on a plane; in this case, the Euler formula is
F - S + V = 1, (1)
where F is the number of faces (minimal polygons of the grid); S is the number of their sides and
V is the number of vertices.
We will calculate the numbers S and V in the formula (1), and then use it to determine F, our major unknown.
Part 1. Calculating the number of vertices V
Let call our triangle ABC, by the names of its vertices.
We have 4 lines from vertex A to the opposite side "a".
We have 4 lines from vertex B to the opposite side "b".
These two families of lines have 4 x 4 = 16 intersection points.
Next, we have 4 lines from vertex A to the opposite side "a",
and we have 4 lines from vertex C to the opposite side "c".
These two families of lines produce other 4 x 4 = 16 intersection points, different from 16 points above.
Finally, we have 4 lines from vertex B to the opposite side "b",
and we have 4 lines from vertex C to the opposite side "c".
These two families of lines produce other 4 x 4 = 16 intersection points, different from 32 points above.
So, we have 16 + 16 + 16 = 48 intersection points INSIDE the triangle ABC.
To it, we should add 5 + 5 + 5 = 15 intersection points along the PERIMETER of the triangle ABC.
In all, there are 48 + 15 = 63 intersection points: V = 63.
Part 2. Calculating the number of sides S
From vertex A to the opposite side "a" we have 4 interior lines inside triangle ABC.
Interesting fact is that each of these lines has 9 elementary interior segments.
You may count and check this fact on your own.
It is not an accidental fact: 9 elementary interior segments in each of these lines are created
by 4 + 4 = 8 intersection points of this line with the family of 4 lines " from B to b " and
with the family of 4 lines " from C to c ".
So, the four lines " from A to a " give us 4*9 = 36 elementary segments.
Similarly, from vertex B to the opposite side "b" we have 4 interior lines inside triangle ABC.
Again, interesting fact is that each of these lines has 9 elementary interior segments.
You may count and check this fact on your own.
The reason is similar to the above case.
So, the four lines " from B to b " give us 4*9 = 36 another elementary segments.
Finally and similarly, we have another 36 elementary segments from the family of lines " from C to c ".
In all, we just counted 36 + 36 + 36 = 108 elementary segments INSIDE the triangle ABC.
Add to it 5 + 5 + 5 = 15 elementary segments along the PERIMETER of triangle ABC.
So, the total number of elementary segments is 108 + 15 = 123: S = 123.
Part 3. Applying the Euler formula
From the formula (1), we have
F = 1 + S - V = 1 + 123 - 63 = 61.
ANSWER. The number of faces inside the triangle ABC (non-overlapping regions) is 61.
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