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Divide ABCDE into two parts with equal area

Albert

Well-known member
Jan 25, 2013
1,225
ABCDE is a pentagon,please construct a line (passing

through point A),and divide ABCDE into two parts with equal

area
 

Prove It

Well-known member
MHB Math Helper
Jan 26, 2012
1,403
Is this a general pentagon, or a particular pentagon?
 

Ackbach

Indicium Physicus
Staff member
Jan 26, 2012
4,191
For a regular pentagon, just construct the midpoint of $CD$, call if $F$, and draw the segment $AF$. This splits the pentagon in two equal pieces.
 

Albert

Well-known member
Jan 25, 2013
1,225

Opalg

MHB Oldtimer
Staff member
Feb 7, 2012
2,703
ABCDE is a pentagon,please construct a line (passing

through point A),and divide ABCDE into two parts with equal

area


Referring to Albert's beautiful solution to the problem in his earlier thread, if $M$ is the midpoint of $PQ$ then the line $AM$ will do the job, provided that $M$ lies between $C$ and $D$. I imagine that this must necessarily be the case, but I don't see how to prove it.
 

Albert

Well-known member
Jan 25, 2013
1,225
In my opinion the best proof is "a proof without words"
so again I construct a diagram and let it explain the solution
pentagon02.jpg
and AG is what we need as written by Opalg
"if [FONT=MathJax_Math]M[/FONT] is the midpoint of [FONT=MathJax_Math]P[/FONT][FONT=MathJax_Math]Q[/FONT] then the line [FONT=MathJax_Math]A[/FONT][FONT=MathJax_Math]M[/FONT] will do the job"
 
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Albert

Well-known member
Jan 25, 2013
1,225
Referring to Albert's beautiful solution to the problem in his earlier thread, if $M$ is the midpoint of $PQ$ then the line $AM$ will do the job, provided that $M$ lies between $C$ and $D$. I imagine that this must necessarily be the case, but I don't see how to prove it.
The statement M lies between C and D is not always true
in fact M and C (or M and D)may coincide
May M also lie between D and E ?(if the length of CD is very small)
M may also lie between B and C.
(we may check this using various diagram)
 
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