# The Joy of Hexasect

This geometric construction challenge is a set of components to be placed in order, $$\begingroup \def \s #1{{ \small\sf #1 }} \def \AB { \overline {\s{AB}} } \def \line #1{{ \small \overleftrightarrow {\s{#1}} }}$$ dividing line segment $$\AB$$ of length 6 into 6 subsegments, each of length 1.

hexasect – transitive verb – to divide into 6 equal parts

How can $$\AB$$ be hexasected by placing 5 circles and 7 lines that produce just 1 lines-only node?

Construction guidelines
• Circles and lines are placed sequentially, in any order that accords with nodes that exist at times of placement.
• A circle may be placed where a node exists for the circle’s center.
• A line may be placed where it crosses at least two existing nodes.
Nodes are endpoints A and B as well as intersections among circles, lines and/or $$\AB$$.
• A lines-only node is an intersection of lines in the completed solution. No circles pass through a lines-only node. (Nodes along $$\AB$$ are ineligible because $$\AB$$ is technically a segment, not a line.)

Example
quadrisect – transitive verb – to divide into 4 equal parts
A different $$\AB$$, of length 4, can be quadrisected by placing 4 circles and 6 lines that produce just 1 lines-only node.

• Step 1 places circles centered at nodes A and B. These circles’ intersections produce nodes C and D.
• Step 2 places line $$\line{BC}$$ and a circle centered at C, whose intersection produces node E, and places $$\line{CD}$$, whose intersection with $$\AB$$ produces node F.
• Step 3 places a circle centered at E. This circle’s intersection with $$\line{BC}$$ produces node G.
• Step 4 places lines $$\line{AD}$$ and $$\line{FG}$$, whose intersection produces node H.
• Step 5 completes this quadrisection by placing lines $$\line{CH}$$ and $$\line{DG}$$. The desired lines-only node is H.

$$\endgroup$$

• This quadrisection example is indeed suboptimal (and more complex than the puzzle's solution, which is tied for optimal hexasection). Optimal quadrisection uses fewer components but doesn't demonstrate a lines-only node.
– humn
Mar 7 at 16:06