File:FS FJCFC2 dia.png
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Captions
Summary
[edit]DescriptionFS FJCFC2 dia.png |
English: Largest circle in a 120° circular sector (fan) that contains the broadest isosceles triangle and the following largest circle and the following largest 120° circular sector (fan)
Deutsch: Größter Kreis in einem Drittelkreis (Fächer), der bereits das breitestes gleichschenkliges Dreieck und den nächstgrößten Kreis sowie den nächstmöglichen größten Drittelkreis enthält |
Date | |
Source | Own work |
Author | Hans G. Oberlack |
0) The 120°-degree circular sector (fan) as base element.
1) Inscribed is the broadest isosceles triangle.
2) Inscribed is the largest circle.
3) Inscribed is the largest 120°-degree circular sector (fan).
4) Inscribed is the next largest circle.
General case
[edit]Segments in the general case
[edit]0) Radius of the base circular sector:
1) side length of the inscribed triangle: , because it is the broadest triangle
2) Radius of the inscribed circle: , see calculation (5)
3) Radius of the inscribed circular sector: , applying calculation (1)
4) Radius of the inscribed circle: , see calculation (8)
Perimeters in the general case
[edit]0) Perimeter of base circular sector:
1) Perimeter of inscribed triangle:
2) Perimeter of inscribed circle around :
3) Perimeter of inscribed circular sector:
4) Perimeter of inscribed circle around :
Areas in the general case
[edit]0) Area of the base circular sector
1) Area of the inscribed triangle , see calculation (3)
2) Area of the inscribed circle around :
3) Area of the inscribed circular sector
4) Area of the inscribed circle around :
Centroids in the general case
[edit]0) By definition the centroid point of a base shape is
1) The centroid of the inscribed triangle relative to the base centroid is: , see Calculation (4)
2) The centroid of the inscribed circle relative to the base centroid is: , see Calculation (6)
3) The centroid of the inscribed circular sector relative to the base centroid is: , see Calculation (7)
4) The centroid of the inscribed circle relative to the base centroid is: , see Calculation (9)
Normalised case
[edit]In the normalised case the area of the base circular sector is set to 1.
So
Segments in the normalised case
[edit]0) Radius of the base circular sector:
1) Side length of the inscribed triangle:
2) Radius of the inscribed circle around :
3) Radius of the inscribed circular sector:
4) Radius of the inscribed circle around :
Perimeter in the normalised case
[edit]0) Perimeter of base circular sector:
1) Perimeter of inscribed triangle:
2) Perimeter of inscribed circle around :
3) Perimeter of inscribed circular sector:
4) Perimeter of inscribed circle around :
S) Sum of perimeters:
Area in the normalised case
[edit]0) Area of the base circular sector is by definition
1) Area of the inscribed triangle
2) Area of the inscribed circle around :
3) Area of the inscribed circular sector
4) Area of the inscribed circle around :
Centroids in the normalised case
[edit]0) Centroid of the base shape:
1) Centroid of the inscribed triangle:
2) Centroid of the inscribed circle around :
3) Centroid of the inscribed circular sector:
4) The centroid of the inscribed circle around :
Calculations
[edit]Given elements
[edit](1)
(2) Angle in M:
(3) Angles in A and B of triangle :, since it is a isosceles triangle
(4) , since the isosceles triangle is symmetric
(5)
(6)
(7) , since is perpendicular to and therefore parallel to
Calculation 1
[edit]Calculating length of MD
, applying equation 3 and the definition of the sinus
, calculating the sine
, applying equation (1)
, rearranging
Calculation 2
[edit]Calculating length of AB
, applying equation 3 and the definition of the cosinus
, calculating the sine
, applying equation (1)
, rearranging
, applying equation (4)
, rearranging
Calculation 3
[edit], calculating the area of triangle
, applying calculation (1)
, applying calculation (2)
, rearranging
Calculation 4
[edit]starting from S_0
, extending to M
, since (0+0i)=0
, applying the centroid formula
, shortening
, calculating the sine
, shortening
, expressing the vector as complex number
, applying calculation (1)
, expressing the vector as complex number
, applying the centroid formular for isosceles triangles
, applying calculation (1)
, shortening
, using distributive property
, adding complex numbers
, adding
, adding
, adding
Calculation 5
[edit]Calculating radius
, applying equation (1)
, applying the construction of the diagram
, applying calculation (1)
, rearranging
, applying equation (5)
, applying equation (5)
Calculation 6
[edit]Calculating the centroid of the circle
, since the centroid of the base shape is (0+0i)=0
, applying equation (5)
, applying equation (1)
, applying calculation (5)
, rearranging
, applying the centroid formula for circular sectors
, rearranging
, calculating the sine
, rearranging
Calculation 7
[edit]Calculating the centroid of the inscribed circular sector
, since the centroid of the base shape is (0+0i)=0
, applying the centroid formular for circular sectors
, rearranging
, calculating the sine
, shortening
, applying the centroid formular for circular sectors
, rearranging
, shortening
, calculating the sine
, shortening
, applying the relation
, calculating
, rearranging
Calculation 8
[edit]Calculating the radius of the inscribed circle around by making use of the red right triangle and the green right triangle
, since the triangles share the side
, since the two circles have the touching point on this line
, replacing by known lenghts
, applying equation (5)
, applying equation (7)
, applying binominal formula
, rearranging
, cancelling out
, rearranging
, applying calculation (5)
, since the circle around is in point G tangential to the circular sector around M
, applying equation (1)
,
,applying equation (6)
,applying equation (7)
,applying calculation(1)
, applying binomial formula
, applying binomial formula
, cancelling out
, rearranging
, rearranging
, rearranging
, reducing
, rearranging
Calculation 9
[edit]Calculating the centroid of the inscribed circle
, since the centroid of the base shape is (0+0i)=0
,
, see calculation (6)
,
,
, see calculation (5)
, see calculation (8)
, rearranging
, rearranging
, rearranging
, rearranging
, rearranging
, applying the Pythagorean theoreme on
,
, applying calculations (5) and (8)
, rearranging
, 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.
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