File:FS RVC2 dia.png

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Original file (1,924 × 1,718 pixels, file size: 84 KB, MIME type: image/png)

Captions

Captions

Second largest circle inscribed in an isosceles right triangle that contains the largest inscribed quarter circle

Summary

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Description
English: Second largest circle inscribed in an isosceles right triangle that contains the largest inscribed quarter circle - Details: RVC2 dia.png
Deutsch: Zweitgrößter eingeschriebener Kreis in einem gleichschenkligen, rechtwinkligen Dreieck, das bereits den größten eingeschriebenen Viertelkreis enthält - Details: RVC2 dia.png
Date
Source Own work
Author Hans G. Oberlack

Shows the second largest quarter circle within a right isosceles triangle that contains the largest quarter circle.

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
Inscribed is the second larges circle with radius and centroid

Segments in the general case

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0) The side length of the base right triangle
1) Radius of the inscribed quarter circle (See calculation 1).
2) Radius of the inscribed second largest circle see Calculation (5).

Perimeters in the general case

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

Areas in the general case

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

Centroids in the general case

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Centroid positions are measured from the centroid point of the base shape
0) Centroid position of the base triangle:
1) Centroid position of the inscribed quarter circle: , (see calculation 4)
2) Centroid position of the inscribed circle: , with , (see calculation 6)

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:
2) Radius of the inscribed circle:

Perimeters in the normalised case

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0) Perimeter of base triangle:
1) Perimeter of the inscribed quarter circle:
2) Perimeter of the inscribed 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
2) Area of the inscribed 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 quarter circle:
2) 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:
0)
1)
2)
S)

Identifying number

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Apart of the base element there are two shapes allocated. Therefore the integer part of the identifying number is 2.
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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Given elements

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Because FS_R is a right isosceles triangle the following equations hold:
(1)
(2)
(3)
(4)
(5) , since both radii are perpendicular to the tangent
(6) , since and are tangent to the circle around in points E and F respectively
(7) , since and are tangent to the circle around in points E and F respectively
(8) , since and are tangent to the circle around in points E and F respectively and therefore the segment divides the angle in B evenly

Calculation 1

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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:








-->

Calculation 5

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Radius of the inscribed circle around
, since is a right triangle
, applying equation (5)
, applying equation (6)
, since
, applying equation (1)
, since is a right triangle
, applying equation (6)
, applying equation (8)
, extending
, applying binomical formula
, multiplying
, multiplying
, applying calculation (1)
, multiplying
, eliminating on both sides of the equation
, subtracting on both sides of the equation
, reducing
, subtracting on both sides of the equation
, rearranging
, multiplying both sides by 2
, integration2 into the brackets
, multiplying
, rearranging
, adding on both sides
, applying binomial formulas
, rearranging
, rearranging
, extracting 4 off the brackets
, applying binomial formula
, extracting the root
, rearranging
, rearranging
, rearranging

Calculation 6

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Centroid position of the inscribed circle relative to the centroid of the base triangle:

, since
, definition of centroid of a triangle
, since applying equation (6)</math>
, rearranging
, rearranging
, rearranging
, rearranging
, applying the Pythagorean theorem on triangle
, applying equation (6)
, applying equation (5)
, applying binomial formula
, rearranging
, applying calculation (1)
, defining as in Calculation (5)
, rearranging
, rearranging
, rearranging
, rearranging
, rearranging
, rearranging
, with

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
current07:33, 8 October 2023Thumbnail for version as of 07:33, 8 October 20231,924 × 1,718 (84 KB)Hans G. Oberlack (talk | contribs)Diagram refined
06:55, 8 October 2023Thumbnail for version as of 06:55, 8 October 20231,924 × 1,710 (84 KB)Hans G. Oberlack (talk | contribs)Uploaded own work with UploadWizard

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