File:FS FJ dia.png
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Captions
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 |
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
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
Licensing edit
- You are free:
- to share – to copy, distribute and transmit the work
- to remix – to adapt the work
- Under the following conditions:
- attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
- share alike – If you remix, transform, or build upon the material, you must distribute your contributions under the same or compatible license as the original.
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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Horizontal resolution | 129.92 dpc |
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Vertical resolution | 129.92 dpc |