Finding the Mid-Point of the side of a square.
Given a square drawn on a piece of paper, is it possible to find the midpoint of one of its sides using the following tools: -A pencil (used ONLY to draw lines) -An infinitely long and infinitesimally thin unmarked straight edge
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$\begingroup$It is possible.
Select a point $P$ outside the square and draw lines through $P$ and the corners $A$ and $B$. These lines intersect $DC$ at the points $E$ and $F$. Now draw the lines $AF$ and $BE$. These lines intersect at $G$. A line through $P$ and $G$ will now intersect $AB$ at its midpoint $M$.
We know $M$ is the midpoint by the following argument:
Using the Intercept Theorem we know that $$ \frac{HF}{MB} = \frac{EH}{AM}$$But from similar triangles we also know that $$ \frac{EH}{MB} = \frac{HF}{AM}$$Combining these two equations we find that $$ \left( \frac {AM}{MB}\right)^2 = 1$$which means $AM = MB$.
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