Level 19: Construct two points, such that the segment is cut into three equal pieces.

Hint: Perhaps unrelated – study the centroid of a triangle.

Solution: The centroid of a triangle is found by intersecting the median, the median is defined by the intersection between the midpoint of an edge and the opposite vertex. The interesting thing about this is that it can be shown that the intersection point always cut the median into two-third on the vertex side and one third on the remaining side. It is this property that we will leverage for this level.

With that in mind, we make AB the median, which means A is the opposite vertex and B is the midpoint of some base line, construct the base line by creating a small circle at B and draw the diameter CD, find the midpoint E of AD and join CE for another median, now we find F which is the centroid of the triangle ACD. That cuts the line in two third/one third, and so we use another circle with center F and radius FB to find the remaining point G.