File:FS FJ dia.png
![File:FS FJ dia.png](https://upload.wikimedia.org/wikipedia/commons/thumb/f/f9/FS_FJ_dia.png/800px-FS_FJ_dia.png?20231025220750)
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Captions
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Summary
[edit]DescriptionFS FJ dia.png |
English: Broadest isosceles triangle in a 120° circular sektor (fan)
Deutsch: Breitestes gleichschenkliges Dreieck in einem Drittelkreis (Fächer) |
Date | |
Source | Own work |
Author | Hans G. Oberlack |
![](https://upload.wikimedia.org/wikipedia/commons/thumb/f/f9/FS_FJ_dia.png/220px-FS_FJ_dia.png)
The 120°-degree circular sector (fan) as base element.
Inscribed is the broadest isosceles triangle.
General case
[edit]Segments in the general case
[edit]0) The radius of the base circular sector:
1) The side length of the inscribed triangle: , because it is the broadest triangle
Perimeters in the general case
[edit]0) Perimeter of base circular sector:
1) Perimeter of inscribed triangle:
Areas in the general case
[edit]0) Area of the base circular sector
1) Area of the inscribed triangle , see calculation (3)
Centroids in the general case
[edit]0) By definition the centroid point of a base shape is
1) The centroid of the inscribed triangle relative to the base centroid is: , see Calculation (4)
Normalised case
[edit]![](https://upload.wikimedia.org/wikipedia/commons/thumb/c/c2/FS_FJ.png/220px-FS_FJ.png)
In the normalised case the area of the base circular sector is set to 1.
So
Segments in the normalised case
[edit]0) Radius of the base circular sector:
1) Side length of the inscribed triangle:
Perimeter in the normalised case
[edit]0) Perimeter of base circular sector:
1) Perimeter of inscribed triangle:
S) Sum of perimeters:
Area in the normalised case
[edit]0) Area of the base circular sector is by definition
1) Area of the inscribed triangle
Centroids in the normalised case
[edit]0) Centroid of the base shape:
1) Centroid of the inscribed triangle:
Calculations
[edit]Given elements
[edit](1)
(2) Angle in M:
(3) Angles in A and B of triangle :
, since it is a isosceles triangle
(4) , since the isosceles triangle is symmetric
Calculation 1
[edit]Calculating length of MD
, applying equation 3 and the definition of the sinus
, calculating the sine
, applying equation (1)
, rearranging
Calculation 2
[edit]Calculating length of AB
, applying equation 3 and the definition of the cosinus
, calculating the sine
, applying equation (1)
, rearranging
, applying equation (4)
, rearranging
Calculation 3
[edit], calculating the area of triangle
, applying calculation (1)
, applying calculation (2)
, rearranging
Calculation 4
[edit]starting from S_0
, extending to M
, since (0+0i)=0
, applying the centroid formula
, shortening
, calculating the sine
, shortening
, expressing the vector as complex number
, applying calculation (1)
, expressing the vector as complex number
, applying the centroid formular for isosceles triangles
, applying calculation (1)
, shortening
, using distributive property
, adding complex numbers
, adding
, adding
, adding
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File history
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Date/Time | Thumbnail | Dimensions | User | Comment | |
---|---|---|---|---|---|
current | 22:07, 25 October 2023 | ![]() | 1,393 × 961 (45 KB) | Hans G. Oberlack (talk | contribs) | Diagram refined |
17:47, 25 October 2023 | ![]() | 1,393 × 835 (42 KB) | Hans G. Oberlack (talk | contribs) | Uploaded own work with UploadWizard |
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