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.
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