File:FS RVC2C2 dia.png
Original file (1,924 × 1,718 pixels, file size: 83 KB, MIME type: image/png)
Captions
Summary edit
DescriptionFS RVC2C2 dia.png |
English: Two second largest circles inscribed in an isosceles right triangle that contains the largest inscribed quarter circle - Details: RVC2C2 dia.png
Deutsch: Zwei zweitgrößte eingeschriebene Kreise in einem gleichschenkligen, rechtwinkligen Dreieck, das bereits den größten eingeschriebenen Viertelkreis enthält - Details: RVC2C2 dia.png |
Date | |
Source | Own work |
Author | Hans G. Oberlack |
Shows the two second largest quarter circle within a right isosceles triangle that contains the largest quarter circle.
Elements edit
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 largest circle with radius and centroid
Inscribed is the next second largest circle with radius and centroid
Segments in the general case edit
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).
3) Radius of the other inscribed second largest circle ( See Calculation 5).
Perimeters in the general case edit
0) Perimeter of base triangle
1) Perimeter of the quarter circle (See calculation 2 )
2) Perimeter of the inscribed circle:
3) Perimeter of the other inscribed circle:
Areas in the general case edit
0) Area of the base triangle
1) Area of the inscribed quarter circle (See calculation 3)
2) Area of inscribed circle
3) Area of other inscribed circle
Centroids in the general case edit
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)
3) Centroid position of the other inscribed circle: , with , for symmetry reasons
Normalised case edit
In the normalised case the area of the base is set to 1.
Segments in the normalised case edit
0) Side length of the base triangle:
1) Radius of the inscribed quarter circle:
2) Radius of the inscribed circle:
3) Radius of the other inscribed circle:
Perimeters in the normalised case edit
0) Perimeter of base triangle:
1) Perimeter of the inscribed quarter circle:
2) Perimeter of the inscribed circle:
3) Perimeter of the other inscribed circle:
S) Sum of perimeters
Areas in the normalised case edit
0) Area of the base triangle
1) Area of the inscribed quarter circle
2) Area of the inscribed circle
3) Area of the other inscribed circle
Centroids in the normalised case edit
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:
3) Centroid positions of the other inscribed circle:
Distances of centroids edit
The distance between the centroid of the base element and the centroid of the circle is:
0)
1)
2)
3)
4)
5)
S)
Identifying number edit
Apart of the base element there are three shapes allocated. Therefore the integer part of the identifying number is 3.
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
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 edit
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 edit
Perimeter of the quarter circle:
Calculation 3 edit
Area of the inscribed quarter circle:
Calculation 4 edit
Centroid of the inscribed circle measured from the centroid of the base triangle:
Calculation 5 edit
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 edit
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 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 | 19:23, 16 October 2023 | 1,924 × 1,718 (83 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 |