# arithmetic – The Pleasure of Hexasect

This geometric development problem is
a set of parts to be positioned so as,
$$begingroup def s #1{{ smallsf #1 }} def AB { overline {s{AB}} } def line #1{{ small overleftrightarrow {s{#1}} }}$$
dividing line phase $$AB$$ of size 6
into 6 subsegments, every of size 1.

hexasect – transitive
verb – to divide into 6 equal elements

How can $$AB$$ be hexasected by inserting 5 circles and seven traces
that produce simply 1 lines-only node?

Development tips

Circles and contours are positioned sequentially,
in any order that accords with nodes that exist at instances of placement.

A circle could also be positioned the place a node exists for the circle’s heart.

A line could also be positioned the place it crosses a minimum of two current nodes.

Nodes are endpoints A and B in addition to
intersections amongst circles, traces and/or $$AB$$.

A lines-only node is an intersection of traces within the accomplished resolution.
No circles go by means of a lines-only node.
(Nodes alongside $$AB$$ are ineligible as a result of $$AB$$
is technically a phase, not a line.)

Instance

verb – to divide into 4 equal elements

A special $$AB$$, of size 4, might be quadrisected
by inserting 4 circles and 6 traces
that produce simply 1 lines-only node.

• Step 1
locations circles centered at nodes A and B.
These circles’ intersections produce nodes C and D.

• Step 2
locations line $$line{BC}$$ and a circle centered at C,
whose intersection produces node E,
and locations $$line{CD}$$,
whose intersection with $$AB$$ produces node F.

• Step 3
locations a circle centered at E.
This circle’s intersection with $$line{BC}$$
produces node G.

• Step 4
locations traces $$line{AD}$$ and $$line{FG}$$,
whose intersection produces node H.

• Step 5
by inserting traces $$line{CH}$$ and $$line{DG}$$.
The specified lines-only node is H.

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