Please help me solve this problem involving a square and it's diagonal

A

Atti0626

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ABCD is a square. An interior point of AB is F and an interior point of AD is E. Draw a perpendicular to the line CE at point E, and a perpendicular to the line CF at point F. The intersection of these perpendiculars is M. Given that the area of the CEF triangle is half the area of the BCDEF pentagon, prove that point M lies on the AC diagonal of the square.
I calculated that if I denote the lenght of DE by x and the lenght of BF by y, and the squares side is 1, the area of the triangle is a(xy)/2. But I don't know how to use this information.
I think I should somehow use the fact that the points on the AC diagonal are an equal distance away from sides AB and AD, and the fact that CFM an CEM are right angled triangles, but I couldn't figure out a use for them.
 
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Can you please show us a drawing for this problem? That way we can follow things more easily and also see if you have the correct drawing. Thanks
 
Jomo said:
What is an interior point? What is an exterior point?
Click to expand...
Well, I can't answer this well in English, but the interior point is a point in the AB and the AD segment. I am not sure what an exterior point means, but I suppose it is a point which is outside of the given segment.
 
Atti0626 said:
Well, I can't answer this well in English, but the interior point is a point in the AB and the AD segment. I am not sure what an exterior point means, but I suppose it is a point which is outside of the given segment.
Click to expand...
Oh, I thought that it said that one point was exterior, but that is not a big deal.
Using your definition of interior point, your points E and F are not correct.
Before you can attempt this problem you must have the correct definition of interior point.
So find the definition in your language and draw a diagram that corresponds to the definition.
 
I don't understand. F is on side AB, E is on side AD, and which exterior point should I draw?
 
Atti0626 said:
I don't understand. F is on side AB, E is on side AD, and which exterior point should I draw?
Click to expand...
I misread your original post and thought that it said exterior point. It was my mistake, sorry.

I can only assume that when you say interior point of AB you mean a point between the two endpoints.
 
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OK, I just look at this problem with my full attention.
Here are some hints.
Label AF = 1-y
Label EA = 1-x
Note, as you did, that the total area is 1
Calculate the area of triangle DEC
Calculate the area of triangle BCF
Calculate the area of triangle EAF
Calculate the area of triangle CEF by subtracting the rest of the area from 1. Clean up result
Calculate the area of BCDEF by subtracting appropriate area from 1.
Now set area of triangle CEF = (1/2)area BCDEF
Look carefully at the last result.
Continue.
PLease report back your results
 
DEC=x/2
BCF=y/2
AEF=(1-x-y+xy)/2
CEF=1/2-xy/2
BCDEF=1/2+x/2+y/2-xy/2
2CEF=1-xy

1-xy=1/2+x/2+y/2-xy/2
1/2=(x+y+xy)/2
x+y+xy=1

I solved for x=y, it is x=sqrt(2)-1
 
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Atti0626 said:
DEC=x/2
BCF=y/2
AEF=(1-x-y+xy)/2
CEF=1/2-xy/2
BCDEF=1/2+x/2+y/2-xy/2
2CEF=1-xy

1-xy=1/2+x/2+y/2-xy/2
1/2=(x+y+xy)/2
x+y+xy=1

I solved for x=y, it is x=sqrt(2)-1
Click to expand...
Your calculations are all good. The last line is not general enough as you were not told that x=y. Solve x+y + xy =1 for x or y. What should you do next? Think.
I need to leave for the next 5 hours but someone else will help you!
 
I calculated that y=(1-x)/(1+x), but don't know what to do with this information. I tried to substitute this value in to the areas, but I couldn't figure out anything with them. How can I arrive at a conclusion about the point M with knowledge about the areas?
 
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