# Divide ABCDE into two parts with equal area

#### Albert

##### Well-known member
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
Is this a general pentagon, or a particular pentagon?

#### Ackbach

##### Indicium Physicus
Staff member
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
Is this a general pentagon, or a particular pentagon?
A general pentagon(convex)

#### Opalg

##### MHB Oldtimer
Staff member
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
In my opinion the best proof is "a proof without words"
so again I construct a diagram and let it explain the solution

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
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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