A known approximate method for constructing a regular heptagon inscribed in a circle of radius $r$ is to use the height $m$ of an equilateral triangle with side $r$ instead of the side of the heptagon. Given that the side of a regular heptagon is $0.867768 r$ to six decimal places, calculate $m$ to six decimal places as well, and then determine how many thousandths of the exact side length the difference between these two values is; this can be considered the accuracy measure, or error, of the given approximate method. - Is the following approximate method better? Let $A$ and $B$ be points on the circle that are $r$ apart, measure a segment of arbitrary length 7 times on the line $AB$ from $A$ towards $B$, connect the reflection of the endpoint $C$ of the seventh segment over the center $O$ of the circle, $D$, to the endpoint $E$ of the second segment, and finally, denoting the intersection of $DE$ with the radius $OA$ by $F$, take $BF$ as the approximate length of the heptagon side.
$m=\sqrt{3} / 2$, to the hundredth $0.866025 r$, so the first method $0.001743 \cdot 1000 / 0.8678=2.01^{0} \% 00^{-k e l ~ h i b a ́ s, ~ a ~ p o n t o s ~}$ value approximates from below.

![](https://cdn.mathpix.com/cropped/2024_05_02_e6ecc5f9ce3cbf6c17d3g-1.jpg?height=532&width=555&top_left_y=295&top_left_x=774)

In the second method, let the midpoint of $DE$ be $G$. Then $OG$ is parallel to $AC$ and $OG=CE / 2=5s / 2$, where $s$ is the length of an arbitrary segment. From the similarity of triangles $FAE$ and $FOG$, and from $OF + FA = r$, we get $OF = 5AF / 4 = 5r / 9$. (Accordingly, the position of $F$ is independent of the choice of $s$, which is expected for the uniqueness of the construction, as we no longer use elements dependent on $s$.) Then, from triangle $OFB$ (or $AFB$) with the angle $\angle BOA = \angle BAO = 60^{\circ}$, using the cosine rule, we get $BF = r \sqrt{61 / 9}$, rounded to six decimal places, $0.867806 r$.

This method gives a larger result than necessary, with an error of $36 \cdot 10^{-6} \cdot 10^{3} / 0.868 < 0.05^{0} /{ }_{00}$, which is less than 1/40th of the previous one, making it better.

George Muszély (Bp. VIII., Vörösmarty M. g. III. o. t.)

Remarks. I. The slight error in the second method justifies calculating to six decimal places. - 2. It is a natural question whether this method was discovered by chance or through deliberate work. It is possible that they arrived at it through the following path. The side length of the heptagon according to the first method is slightly larger than $BH$, where $H$ is the midpoint of the radius $OA$ (while according to the second method, it is smaller than $BF$, but this was not known at the beginning of the research). Where would the segment $HA$ be intersected by a circle centered at $B$ with a radius equal to the exact side length of the heptagon? Denoting this point by $J$, we have $HJ^2 = BJ^2 - BH^2 = 0.867768^2 r^2 - 0.75 r^2 = (0.753021 - 0.75) r^2 = 0.003021 r^2, HJ = 0.05496 r, OJ = 0.55496 r$. The coefficient is close to $0.555 \ldots = 5 / 9$, which might have given the idea for the method. The significant role of point $B$ in the method; in the preparation, any other line could have been used instead of $BA$.

The construction chosen to mark the point $F$ that divides $OA$ in the ratio $5:4$ is also interesting. By transitioning from $C$ to $D$, we can avoid the usual parallel construction.