H E P T A G O N

By Robin Hu

**Step 1:** Draw an arbitrary circle centered at `O`. Draw `OA`, the radius of circle `O`.

**Step 2:** Draw a circle centered at `A` with a radius of length `AO`. Circle `A` intersects the original

circle at points `B` and `C`.

**Step 3:** Draw a line through `BC`.Lines `OA` and `BC` intersect at `D`.

**Step 4:** Draw a circle centered at `B` with a radius of `BD`. This circle intersects circle `O` at `E`.

**Step 5:** `BE` is a side of the Heptagon, use it to find the others.

Out of all the constructions, in my opinion, this first one is the most easiest and most accurate. Amazingly, this construction can only be found on My website. If you bisect one of the sides, you can create an approximation for a 14-gon.

**Step 1:** Draw a circle `A`. Draw segments `AB` and `AC` so that `AB` is perpendicular to `AC`.

Draw a line through `BC`.

**Step 2:** Draw line `BD` so that `BC` is perpendicular to `BD` and `BD`
is one half of `BC`.

**Step 3:** Draw a line through `DC`. Draw a circle at `D` with radius `DB`. Circle `D`
intersects `DC` at `E`.

**Step 4:** Draw a circle at point `C` with radius `CE`. This circle intersects the original circle at two points
of the heptagon.

This construction on the other hand, was actually invented by me. I found it by taking the Golden Mean of a side of square.
Quite accurate for an amatuer construction.

**Step 1:** Draw circle `O` with diameter `AB` drawn across it.

**Step 2:** Divide `AB` into `7` equal parts. Call the **second** point from the left `P`.

**Step 3:** Draw two circles, one centered at `A` and one centered at `B`.Both circles should have a
radius of diameter `AB`. The two circles intersect at `C`.

**Step 4:** Draw a line through `CP`. Line `CP` intersects circle `O` at `D`.
`AD` is a side of the heptagon.

This construction is actually based on the construction of an n-gon.

**Step 1:** Draw circle `O`. Inscribe a pentagon `ABCDE` inside circle `O`.

**Step 2:** Draw a circle inside of the pentagon, tangent to all sides of the pentagon (Find the midpoint `M` of
`AB` and draw a circle at `O` with radius `OM`).

**Step 3:** Draw a line through `OA`. `OA` intersects the tangent circle at `F`. Draw a circle at `F`
with radius `FA`. Circle `F` intersects line `OA` at `H`.

**Step 4:** Draw a circle at `A` with a radius of `AH` or copy the length of `AF` twice over line
`OA`. `AG` is twice the length of `AF`.

**Step 5:** Draw a circle at `O` with radius `OH`. Inscribe an equilateral triangle `HIJ`
within this circle. Draw a line through `IJ`.

**Step 6:** Draw a circle at `O` with a radius of `OG`. This large circle intersects line `IJ`
at points `P` and `P2`.

**Step 7:** Points `G,P, and P2` are points of the heptagon.

See the NexusJournal website for more details.

NEW!!!

I have included a new, and totally different construction for the heptagon. It involves the use of an equally spaced x-y grid.

It's very simple. Draw a circle, with the center (0,0) and intersecting the point (2,4) and (-2,4) on the grid. Make one of the points of the heptagon the point where the circle intersects the y-axis. Then, where the line y= -1 intersects the circle, draw two more points. There are now three points of the heptagon. You can now find the other 4 points easily.

Construction | Angle Measure | Percent Error |
---|---|---|

Regular Heptagon | 51.428571° | 0.000% |

Construction #1 | 51.317812° | 0.215% |

Construction #2 | 51.827292° | 0.775% |

Construction #3 | 51.518222° | 0.174% |

Construction #4 | 51.460483° | 0.062% |

Construction #5 | 51.4605° | 0.062% |

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Email me if you have questions or comments.

Square | Triangle and Hexagon | Pentagon | Golden Ratio | Nonagon |

Pentadecagon | Heptadecagon | N-Gon | Squaring a Circle | Basic Constructions |

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Visitor Mark | Date Passed |
---|---|

800 | 10/20/2002 |

1000 | 11/1/02 |

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