Doubling the Square
The Problem.
Given a square, construct a square with double the area using straight edge
alone.
----------------------------------------This is easy using a straight edge and compasses but this problem asks for a
construction using only a straight edge.
This implies that you cannot measure a length and transfer this length, nor
can you slide the straight edge to create parallel lines; all you can do is use
the straight edge to draw a line between two given points.
The diagram below shows that if you could bisect a line segment using a
straight edge alone then, having bisected the edges and extended the four
edges of the square, you can then construct the large square shown which
clearly has double the area of the square ABCD .
But how would you bisect a side of the square with only a straight edge?
Or would you use a different method altogether?
© MEI 2009
Method 1 Try this Geogebra file.
The diagram shows a line segment PQ and a
line parallel to PQ .
First choose any point R as shown and
construct lines PR and QR . These lines
intersect the parallel line at A and B .
Now construct lines PB and QA and call the
point of intersection C .
Finally draw line RC which cuts PQ at E .
We will prove that E is the midpoint of PQ .
One way to do this, although perhaps not the most elegant, is to look for
PQ
similar triangles, giving two expressions for the ratio
.
AB
Triangles PQR and ABR are similar

PQ PR

AB AR
1
Triangles PER and ADR are similar

PR PE

AR AD
 2
PQ PE

AB AD
 3
Combining 1 and  2  gives
Now think about these similar triangles
EQ QC

DA AC
PQ QC

BA AC
 4
Combining  4  and  5  gives
Finally  3 and  6  give
PQ EQ

BA DA
 5
 6
EQ PE

and so E is indeed the midpoint of PQ .
DA AD
© MEI 2009
Method 2 Try this Geogebra file.
In the construction below the point P has been chosen on line BD and gives
points S and T. Looking at this another way, you could have chosen the point
S and used this to locate P and then T. The important point to note is that the
line segment TU is parallel and equal in length to AC.
You can repeat this process four times, choosing your new starting points
carefully as shown in the Geogebra file, to construct a square with double the
area of ABCD.
Method 3 Try this Geogebra file.
Starting with square ABCD with centre E, choose an arbitrary point F on AD
then join pairs of points to create additional points (in alphabetical order).
Prove that square BMLD has twice the area of square ABCD.
© MEI 2009