Geometry: using only compasses and straightedge
Find the centre of a circle
If you drew a circle with your compasses you would know where the centre is: the compasses would leave a mark at the centre. You could fold the paper in half and then in half again, and when you unfold the centre would be where the two creases meet.
How did the ancient mathematicians do it? Or the monks who drew perfect circles in beautiful manuscripts? Or ancient architects? The Egyptians who built the pyramids knew, and the ancient Greeks who built amazing temples.
Here is how they found the centre of a circle.
Draw a circle with you compasses.
Draw two straight lines AB and CD touching the edge of the circle.
Now we will draw a perpendicular through the centre of one line. Set the compasses at any width bigger than half the line AB. Draw two circles centred at A and B.
Now draw a line through the two points where the circles touch. Long ago Euclid proved that this cuts the line in half at a right angle.
In the same way draw a perpendicular to the mid point of CD.
Where the two perpendiculars meet (red dot) is the centre of the circle.
Why is this so? The lines are parts of the diameter of the circle. It is just like when you folded a paper circle in half. Where the two creases (diameters) meet is the centre.