Divide a line in half (bisect)using a straightedge and compasses.
The straightedge is not a ruler - you cannot measure the line and half it with arithmetic.
The straightedge is not a ruler - you cannot measure the line and half it with arithmetic.
Draw a line with your straightedge. Call the ends A and B.
Use your compasses to draw a circle using A as the center.
Make the compasses wider than half the line. (Roughly guess the width to be more than half.)
Make the compasses wider than half the line. (Roughly guess the width to be more than half.)
Draw another circle around B keeping the compasses set at the same width. The circles cross at C and D.
Draw a straight line with your staightedge from C to D. It crosses the line AB at half way along its length, point E. Success!
Draw a straight line with your staightedge from C to D. It crosses the line AB at half way along its length, point E. Success!
Join ACBD to make a quadrilateral.
AC, AD are radii of the same circe, so are the same length. BC and BD are the same length too. And when you drew the two circles you used the same size on your compasses, so all these sides are the same length.
Centuries ago a famous mathematician called Euclid proved that a quadrilateral with the same length sides is a parallelogram. And he proved that the diagonals across a parallelogram cut each other in half.
ACBD is a paralellogram
AB and CD are its diagonals, so they cut each other in half. So AE = EB.
AC, AD are radii of the same circe, so are the same length. BC and BD are the same length too. And when you drew the two circles you used the same size on your compasses, so all these sides are the same length.
Centuries ago a famous mathematician called Euclid proved that a quadrilateral with the same length sides is a parallelogram. And he proved that the diagonals across a parallelogram cut each other in half.
ACBD is a paralellogram
AB and CD are its diagonals, so they cut each other in half. So AE = EB.
Geometry: using only compasses and straightedge