File:FS RV dia.png

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Summary

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Description
English: Largest quarter circle in a rectangular isosceles triangle
Deutsch: Größer Viertelkreis in einem gleichschenkligen rechtwinkligen Dreieck
Date
Source Own work
Author Hans G. Oberlack

Shows the largest quarter circle within a right isosceles triangle.

Elements

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Base is the right isosceles triangle of side length and centroid

Inscribed is the largest possible quarter circle with radius and centroid

General case

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Segments in the general case

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0) The side length of the base right triangle
1) Radius of the quarter circle (See calculation 1).

Perimeters in the general case

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0) Perimeter of base triangle
1) Perimeter of the quarter circle (See calculation 2 )

Areas in the general case

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0) Area of the base triangle
1) Area of the inscribed circle (See calculation 3)

Centroids in the general case

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Centroid positions are measured from the centroid point of the base shape
0) Centroid positions of the base triangle:
1) Centroid positions of the inscribed quarter circle: , (see calculation 4)

Normalised case

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Black-and-White version

In the normalised case the area of the base is set to 1.

Segments in the normalised case

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0) Side length of the base triangle
1) Radius of the inscribed quarter circle

Perimeters in the normalised case

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0) Perimeter of base triangle

1) Perimeter of the inscribed quarter circle
S) Sum of perimeters

Areas in the normalised case

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0) Area of the base triangle
1) Area of the inscribed quarter circle

Centroids in the normalised case

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Centroid positions are measured from the centroid point of the base shape.
0) Centroid positions of the base triangle:
1) Centroid positions of the inscribed circle:

Distances of centroids

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The distance between the centroid of the base element and the centroid of the circle is:

Identifying number

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Apart of the base element there is only one shape allocated. Therefore the integer part of the identifying number is 1.
The decimal part of the identifying number is the decimal part of the sum of the perimeters and the distances of the centroids in the normalised case.



So the identifying number is:

Calculations

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Calculation 1

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Because FS_R is a right isosceles triangle the following equations hold:
(1)
(2)
(3)
(4)

Since is tangent to the quarter circle in point the triangle has an right angle in . This means:
(5)
, applying equations (1),(3) and (4)


Calculation 2

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Perimeter of the quarter circle:






Calculation 3

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Area of the inscribed quarter circle:






Calculation 4

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Centroid of the inscribed circle measured from the centroid of the base triangle:








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Licensing

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I, the copyright holder of this work, hereby publish it under the following license:
w:en:Creative Commons
attribution share alike
This file is licensed under the Creative Commons Attribution-Share Alike 4.0 International license.
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/TimeThumbnailDimensionsUserComment
current19:09, 12 March 2022Thumbnail for version as of 19:09, 12 March 2022687 × 601 (20 KB)Hans G. Oberlack (talk | contribs)upload corrected
17:55, 12 March 2022Thumbnail for version as of 17:55, 12 March 2022687 × 601 (20 KB)Hans G. Oberlack (talk | contribs)Uploaded own work with UploadWizard

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