Question

Let ABC a triangle. Take n points on the side AB, join them with C by straight lines, take n points on BC, join them with A by straight lines, take n points on AC, join them with B by straight lines. If no three straight lines meet at any point except at A,B,C then find the number of partitions in the triangle.
Please show the solution with proof

Answer 1

@Callisto

Answer 2

(1) When you draw a line from a vertex to the opposite side, each time you meet another line you create one more region.

Initially there is 1 region (the original triangle)

When you draw a line from C to AB, you meet one line (the side AC) so each line creates one more region.

n lines create n+1 regions altogether (so far, the result is obvious)

(2) When you draw a line from B to AC, you meet n+1 lines (the n lines you drew in step 1 and the side AC. So each line create another n+1 regions

Adding a lines create n(n+1) regions, plus the original n+1, = (n+1)^2

(3) When you draw a line from A to BC you meet 2n+1 lines (the n lines from B, the n lines form C, and the side BC)

Total number of regions = n(2n+1) + (n+1)^2
= 3n*2 + 3n + 1

http://in.answers.yahoo.com/question/index?qid=20100210065419AAbX9uq

Answer 3

@Arnab09 do u get it

Answer 4

I am not sure about the first statement. If it is true, then its alright :)

Answer 5

sorry, I posted this problem in physics section by mistake :(