File:FS CRCC2 dia.png
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Captions
Summary
[edit]DescriptionFS CRCC2 dia.png |
English: Second largest circle adjacent to largest circle in a circle with the largest isosceles triangle
Deutsch: Zweitgrößter Kreis neben dem größten Kreis in einem Kreis mit dem größten gleichschenkligen Dreieck |
Date | |
Source | Own work |
Author | Hans G. Oberlack |
Elements
[edit]Base is the circle of given radius around point
Inscribed is the largest possible right triangle with side length .
Added is the largest circle around point
Added is the second largest circle around point
General case
[edit]Segments in the general case
[edit]0) The radius of the base circle
1) Side of the triangle , see CRC dia
2) The radius of the inscribed circle
3) The radius of the second inscribed circle ,see Calculation 1
Perimeters in the general case
[edit]0) Perimeter of base circle
1) Perimeter of the triangle
2) Perimeter of inscribed circle
3) Perimeter of second inscribed circle
Areas in the general case
[edit]0) Area of the base circle
1) Area of the inscribed triangle
2) Area of the inscribed circle
3) Area of the second inscribed circle
Covered surface of the base shape
Centroids in the general case
[edit]1) Centroids as graphically displayed
[edit]0) Centroid position of the base circle:
1) Centroid positions of the inscribed triangle: , see CR dia
2) Centroid positions of the inscribed circle: , see CRC dia
3) Centroid positions of the second inscribed circle: , see Calculation 2
W) Weighted average centroid: , see Calculation 3
2) Orientated centroids
[edit]The centroid positions of the following shapes will be expressed orientated so that the first shape n with will be of type with . This means that the graphical representation will not correspond to the mathematical expression.
0) Orientated centroid position of the base circle:
1) Orientated centroid position of the inscribed triangle:
2) Orientated centroid position of the inscribed circle:
3) Orientated centroid position of the inscribed circle:
Normalised case
[edit]In the normalised case the area of the base is set to 1.
Segments in the normalised case
[edit]0) Radius of the base circle
1) Side length of the inscribed triangle
2) Radius of the inscribed circle
3) Radius of the second inscribed circle
Perimeters in the normalised case
[edit]0) Perimeter of base circle
1) Perimeter of the inscribed triangle
2) Perimeter of inscribed circle:
3) Perimeter of second inscribed circle
S) Sum of perimeters
Areas in the normalised case
[edit]0) Area of the base circle
1) Area of the inscribed triangle
2) Area of the inscribed circle
3) Area of the second inscribed circle
Centroids in the normalised case
[edit]1) Centroids as graphically displayed
[edit]Centroid positions are measured from the centroid point of the base shape.
0) Centroid position of the base circle:
1) Centroid position of the inscribed triangle:
2) Centroid position of the inscribed circle:
3) Centroid position of the second inscribed circle:
W) Weighted average centroid:
2) Orientated centroids
[edit]The centroid positions of the following shapes will be expressed orientated so that the first shape n with will be of type with . This means that the graphical representation will not correspond to the mathematical expression.
0) Orientated centroid position of the base circle:
1) Orientated centroid position of the inscribed triangle:
2) Orientated centroid position of the inscribed circle:
3) Orientated centroid position of the inscribed circle:
Calculations
[edit]Calculation 1
[edit](1) is perpendicular on since the circle around is tangent to
and is perpendicular on since the circle around is tangent to
and
(2) is perpendicular in since the circle around is tangent to the circle around with and
(3) is perpendicular in since the circle around is tangent to the circle around with and
(4)
, applying eqaution (1)
(5) , applying the Pythagorean theorem
, applying equation (1)
, applying equation (2)
, applying binomial theorem
(6) , applying the Pythagorean theorem
, applying equation (1)
, applying equation (5)
, applying equation (4)
, applying equation (3)
, applying binomial theorem
, applying binomial theorem
, deducting
, adding
, adding
, since
Calculation 2
[edit]The centroid can be written as
, since
expressed as vectors
using vectors of known length
since is on the 45-degree axis
since is perpendicular to
since and
since
since
since
since
since
Calculation 3
[edit]
-->
Calculation 4
[edit]Calculating in the orientated position:
, see calculation 1
, applying Pythagorean theoreme
, applying calculation 1
, applying calculation 1
, applying binomial formula
, eliminating
, since
, rearranging
, rearranging
, rearranging
, rearranging
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 | 17:17, 24 April 2022 | 589 × 648 (35 KB) | Hans G. Oberlack (talk | contribs) | Uploaded own work with UploadWizard |
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