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![[parent]](http://images.planetmath.org:8080/images/uparrow.png) Viewing Correction to 'equilateral triangle'
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more general by Wkbj79
Correction id: 12330
Filed on: 2007-06-06 00:44:45
Status: Rejected on 2007-06-26 17:46:44
Type: Meta/Minor
Correction text:
Quite a few of the statements after the picture can be generalized. For the purpose of parallelism, I suggest the following:
By the theorem at angles of an isosceles triangle, we can conclude that, in any geometry in which SAS holds, an equilateral triangle is regular. In any geometry in which ASA, SAS, SSS, and AAS all hold, the isosceles triangle theorem yields that the bisector of any angle of an equiangular triangle coincides with the height, the median and the perpendicular bisector of the opposite side.
The following statements hold in Euclidean geometry for an equilateral triangle.
\begin{itemize}
\item The triangle is determined by specifying one side.
\item If $r$ is the length of the side, then the height is equal to $\displaystyle \frac{r\sqrt{3}}{2}$.
\item If $r$ is the length of the side, then the area is equal to $\displaystyle \frac{r^2\sqrt{3}}{4}$.
\end{itemize} |
Comment from object owner Mathprof:
Quite a few of the statements after the picture can be generalized. For the purpose of parallelism, I suggest the following:
By the theorem at angles of an isosceles triangle, we can conclude that, in any geometry in which SAS holds, an equilateral triangle is regular. In any geometry in which ASA, SAS, SSS, and AAS all hold, the isosceles triangle theorem yields that the bisector of any angle of an equiangular triangle coincides with the height, the median and the perpendicular bisector of the opposite side.
The following statements hold in Euclidean geometry for an equilateral triangle.
\begin{itemize}
\item The triangle is determined by specifying one side.
\item If $r$ is the length of the side, then the height is equal to $\displaystyle \frac{r\sqrt{3}}{2}$.
\item If $r$ is the length of the side, then the area is equal to $\displaystyle \frac{r^2\sqrt{3}}{4}$.
\end{itemize}
I like to limit the entry to the definition and a few simple properties.
If you want to create an entry, say, some theorems about an equilateral triangle go ahead. but i would like to see it say things like
in a neutral geometry or hyperbolic geometry, rather than in term of specific axioms or theorems such as SSS or ASA . Maybe even ordered geometry if that is sufficient. |
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