Given arbitrary points , can we construct (only using compass and straightedge) an equilateral triangle such that lie on the sides of ?
Yes, we can! Try it for yourself before reading on.
We use the following sequence of steps:
Draw a arbitrary line from point not touching point and
Take an arbitrary point on the line and draw another line from till such that (see how to draw with compass here)
Extend a line from till such that (see how to draw)
Extend intersecting at
As in step draw angle
As in step join and
As in step Draw line
Extend and and let them intersect at
Now points and lie on the sides of the equilateral triangle
After step As from step Similarly we can prove that From angle sum property of triangle From and it is proved that is an equilateral triangle
Now a bonus : Given arbitrary points and can you draw a square such that points and lies on the sides - A problem given by Jeff Giff