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I remember being presented a mathematical puzzle some years back that I still can't solve. The problem is defined as follows:

We have two points on a plane, and using only a compass, how do we find other two points, so that all four of them would be vertices of a square?

I'm not sure whether the first two points were supposed to be vertices of the same edge of a square or not, so solutions to both variants are welcome.

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The trick is to figure out how to construct a pair of points which $\sqrt{2}$ apart... –  Zhen Lin Mar 5 '12 at 7:46
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1 Answer

Here is a description of one of the constructions:

Given the two points $\color{maroon}0$ and $\color{maroon}d$ that form one side of the square:

  1. Construct the five $\color{gray}{\text{gray}}$ circles of common radius $\color{maroon}{od}$ shown in the diagram and locate the points $\color{maroon}a$, $\color{maroon}b$, and $\color{maroon}c$.
  2. Construct the two $\color{maroon}{\text{maroon}}$ circles: one with center $\color{maroon}c$ and radius $\color{maroon}{cb}$, and one with center $\color{maroon}d$ and radius $\color{maroon}{ad}$. Note that $\color{maroon}{ad}$ and $\color{maroon}{cb}$ have the same length.
  3. Locate the point of intersection $\color{pink}e$ of the two maroon circles.
  4. Construct the $\color{pink}{\text{pink}}$ circles of radius $\color{pink}{oe}$ at centers $\color{maroon}c$ and $\color{maroon}d$.
  5. The point of intersection, $\color{pink}f$, of the pink circles is a vertex of the square.
  6. Draw the $\color{darkgreen}{darkgreen}$ circle centered at $\color{pink}f$ of radius $\color{maroon}{od}$.
  7. The point of intersection, $\color{darkgreen}g$, of the darkgreen circle with the gray circle centered at $\color{maroon}d$ is the final vertex of the square.

    enter image description here

Justification of step 5:

From step 1., $\color{maroon}{aboc}$ is a rhombus with common side length $ l(\color{maroon}{co})$. Since the point $\color{pink}e$ is equidistant to the points $\color{maroon}c$ and $\color{maroon}d$ , the segment $\color{pink}e\color{maroon}o$ is perpendicular to the segment $\color{maroon}{cd}$. Since $\color{pink}f$ is equidistant to the points $\color{maroon}c$ and $\color{maroon}d$, the points $\color{pink}e$, $\color{pink}f$, and $\color{maroon}o$ are colinear and the segment $of$ is perpendicular to the segment $\color{maroon}od$.

We need to show that $l(fo)=l(co)$. Proceeding with some abuse of notation:

Considering the rhombus $aboc$, we have $$ cb^2+ao^2=2(ac^2+co^2 ); $$ or, since $ao=co=ac$, $$ cb^2=3co^2. $$ Since $cb=ce$, we have $$\tag{1} ce^2=3co^2 $$
Considering now the right triangle $ceo$: $$\tag{2} ce^2=eo^2+co^2. $$ Combining equations $(1)$ and $(2)$: $$\tag{3} eo^2=2co^2. $$

Considering now the right triangle $cfo$: $$\tag{4} cf^2=fo^2+co^2 $$ since $cf=oe$: $$\tag{5} oe^2=fo^2+co^2 $$ From $(3)$ and $(5)$ now, we finally obtain $$ fo=co, $$ as desired.

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