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# Triangle

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+2493

excuse me for writings

Solveit  Jan 12, 2016

#1
+78719
+10

Note that triangle FBE ≈ triangle ABC

And similar triangles are similar in all respects

And since  BE/EC  = 1/2, then BE/BC   = [ BE] / [ BE + EC]  =  1 / [1 +2] = 1/3

Thus, the base of triangle FBE    = 1/3  the base of  triangle ABC

So......the height of triangle FBE will also be 1/3 of that of triangle ABC

This means that  the area of traingle FBE = (1/3)^2 area of triangle ABC  = 1/9 the area of triangle ABC

And  triangle  DEC  ≈ triangle ABC

And since DC = 2/3 of AC, by similar reasoning as above......each of its dimensions will be 2/3 of those of triangle ABC

So.......the area of triangle DEC = (2/3)^2 the area of triangle ABC = 4/9 the area of triangle ABC

Thus.....area of  triangle ABC - area of  triangle FBE - area of  triangle DEC  = the area of AFED

So

ABC - (1/9) ABC - (4/9) ABC  =  ABC - (5/9)ABC  =  (4/9)ABC

Thus, AFED  is  (4/9) the area  of triangle ABC

CPhill  Jan 12, 2016
Sort:

#1
+78719
+10

Note that triangle FBE ≈ triangle ABC

And similar triangles are similar in all respects

And since  BE/EC  = 1/2, then BE/BC   = [ BE] / [ BE + EC]  =  1 / [1 +2] = 1/3

Thus, the base of triangle FBE    = 1/3  the base of  triangle ABC

So......the height of triangle FBE will also be 1/3 of that of triangle ABC

This means that  the area of traingle FBE = (1/3)^2 area of triangle ABC  = 1/9 the area of triangle ABC

And  triangle  DEC  ≈ triangle ABC

And since DC = 2/3 of AC, by similar reasoning as above......each of its dimensions will be 2/3 of those of triangle ABC

So.......the area of triangle DEC = (2/3)^2 the area of triangle ABC = 4/9 the area of triangle ABC

Thus.....area of  triangle ABC - area of  triangle FBE - area of  triangle DEC  = the area of AFED

So

ABC - (1/9) ABC - (4/9) ABC  =  ABC - (5/9)ABC  =  (4/9)ABC

Thus, AFED  is  (4/9) the area  of triangle ABC

CPhill  Jan 12, 2016
#2
+2493
+5

Thanks CPhill ! :)

Solveit  Jan 12, 2016

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