In a convex-decagon, three diagonals are concurrent. From this information find the number of points at which the diagonals intersect with each other. Also find into how many regions is the polygon divided into by the diagonals?(adsbygoogle = window.adsbygoogle || []).push({});

At the first attempt to solve the question I tried with finding the number of diagonals. There are ten different point and each point can be joined with 9 other points. In this the segment joining the adjacent sides are the sides of polygon itself. Therefore the number of diagonal are 70 as they are 7 from each point. But the actual number should be 35, since in the found 70, for example AD and DA are different diagonals.

But now three diagonals, infact it can be assumed that only three diagonals are concurrent. The point of intersections of all other diagonals are distinct. I cannot frame a decagon to say at how many points should the diagonals intersect. Let the polygon is ABCDEFGHIJ, then diagonal like AD, GJ have no chance of intersecting with each other. But I cannot frame out which all diagonal can never intersect with each other and which all will.

Then started with small steps like diagonals AC, BD, CE, DF, EG, FH, GI, HJ, IA and JB will intersect at ten different points. The decagon if regular, will be divided into 16 regions. However I am not sure the number remains same even in the following condition also

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# Find the number of points at which the diagonals intersect

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