File:FS QJC2 dia.png
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Contents
Summary
[edit]DescriptionFS QJC2 dia.png |
English: Second largest circle in a square containing an inscribed isosceles triangle
Deutsch: Zweitgrößte Kreis in einem Quadrat, das ein eingeschriebenes gleichschenkliges Dreieck enthält |
Date | |
Source | Own work |
Author | Hans G. Oberlack |
![](https://upload.wikimedia.org/wikipedia/commons/thumb/4/46/FS_QJC2_dia.png/220px-FS_QJC2_dia.png)
Task
[edit]The second largest circle inscribed in a square of side length , that contains the largest non-right isosceles triangle
General case
[edit]Segments in the general case
[edit]0) The side length of the base square:
1) Side length of the inscribed isosceles triangle: , see Calculation 1
2) Radius of the inscribed square: , see Calculation 4
Perimeters in the general case
[edit]0) Perimeter of base square:
1) Perimeter of the inscribed triangle:
2) Perimeter of the inscribed circle:
S) Sum of perimeters:
Areas in the general case
[edit]0) Area of the base square:
1) Area of the inscribed triangle:
2) Area of the inscribed circle:
Centroids in the general case
[edit]0) Centroid position of the base square:
1) Centroid position of the inscribed triangle measured from the centroid of the base shape: , see calculation 2
2) Centroid position of the inscribed circle measured from the centroid of the base shape: , see Calculation (5)
W) Weighted centroid: , see calculation 3
Normalised case
[edit]![](https://upload.wikimedia.org/wikipedia/commons/thumb/0/05/FS_QJC2.png/220px-FS_QJC2.png)
In the normalised case the area of the square is set to 1.
So
Segments in the normalised case
[edit]0) Side lenght of the base square
1) Side length of the inscribed triangle:
2) Radius of the inscribed square:
Perimeters in the normalised case
[edit]0) Perimeter of base square
1) Perimeter of the inscribed triangle:
2) Perimeter of the inscribed circle:
S) Sum of perimeters:
Areas in the normalised case
[edit]0) Area of the base square is by definition
1) Area of the inscribed triangle:
2) Area of the inscribed circle:
Centroids in the normalised case
[edit]0) Centroid position of the base square:
1) Centroid position of the inscribed triangle:
2) Centroid position of the inscribed circle measured from the centroid of the base shape:
W) Weighted centroid:
Distances of centroids
[edit]The distance between the centroid of the base element and the centroid of the triangle is:
Sum of distances:
Identifying number
[edit]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
[edit]Given elements
[edit](1) , since
is a square
(2)
(3)
(4)
Calculation 1
[edit], applying the Pythagorean theorem on triangle
, applying equation (2)
, applying equation (1)
, applying equation (3)
, multiplying
, multiplying
,
Calculation 2
[edit]
, applying equation (2)
, applying the formula for centroids of triangles
, applying equation (1)
, rearranging
, rearranging
, rearranging
Calculation 3
[edit]
, since
, since
, since
, eliminating
, rearranging</math>
, eliminating 8
, rearranging
, since
, since
, rearranging
, rearranging
, rearranging
, rearranging
, rearranging
, rearranging
Calculation 4
[edit], applying equation (3)
, applying the result of calculation (1)
, since
and
emanate from point
and are tangent to the circle around
with radius
,since
, applying equation (2)
, applying equation (4)
,since
and
emanate from point
and are tangent to the circle around
with radius
, since
, applying equation (1)
, applying equation (4)
, rearranging
, rearranging
, rearranging
, rearranging
Calculation 5
[edit]
, applying equation (2)
, applying equation (2)
, applying equation (4)
, applying equation (4)
, rearranging
, rearranging
, applying the result of Calculation (4)
, rearranging
, rearranging
, rearranging
Licensing
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- to share – to copy, distribute and transmit the work
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- Under the following conditions:
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File history
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Date/Time | Thumbnail | Dimensions | User | Comment | |
---|---|---|---|---|---|
current | 22:56, 28 October 2022 | ![]() | 645 × 601 (21 KB) | Hans G. Oberlack (talk | contribs) | Uploaded own work with UploadWizard |
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