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