MHB Inscribed and circumscribed quadrilateral

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The discussion revolves around a problem involving a quadrilateral ABCD that is both inscribed in and circumscribed around circles, with a specific relationship between its area and that of another quadrilateral KLMN formed by tangent points. Participants express uncertainty about how to approach the problem, particularly regarding the significance of the area notation S_{ABCD} and the geometric properties of ABCD. One contributor suggests that the quadrilateral may be a trapezium due to symmetry. The term "bicentric" is introduced, indicating that the quadrilateral has both inscribed and circumscribed circles. The conversation highlights the need for further exploration of the properties and relationships of the quadrilateral to solve the problem.
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I would like to discuss the following problem.

The quadrilateral $$ABCD$$ is inscribed into a circle of given radius $$R.$$ And it is circumscribed to a circle. The tangent points from the second circle produce another quadrilateral $$KLMN$$ such that $$S_{ABCD}=3S_{KLMN}.$$ Also $$\gamma$$ is the angle between diagonals $$AC$$ and $$BD.$$ Find the area of $$ABCD.$$

I have no ideas. I wonder if I have to search any regularities of $$ABCD.$$ All given elements seem to me "distanced" from each other.
 
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Andrei said:
I would like to discuss the following problem.

The quadrilateral $$ABCD$$ is inscribed into a circle of given radius $$R.$$ And it is circumscribed to a circle. The tangent points from the second circle produce another quadrilateral $$KLMN$$ such that $$S_{ABCD}=3S_{KLMN}.$$ Also $$\gamma$$ is the angle between diagonals $$AC$$ and $$BD.$$ Find the area of $$ABCD.$$

I have no ideas. I wonder if I have to search any regularities of $$ABCD.$$ All given elements seem to me "distanced" from each other.

I can't give you a complete solution, sorry, but...

1. For symmetry reasons I assumed that the quadrilateral in question must be a trapezium.

2. The diagonals of ABCD and KLMN intersect in the same point.

3. Since I don't know what $$S_{ABCD}$$ means I can't give you any calculations.
 

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earboth said:
1. For symmetry reasons I assumed that the quadrilateral in question must be a trapezium.
...
3. ... I don't know what $$S_{ABCD}$$ means ...
The red quadrilateral in your picture can also be circumscribed. $$S$$ is the area.
 
Andrei said:
I would like to discuss the following problem.

The quadrilateral $$ABCD$$ is inscribed into a circle of given radius $$R.$$ And it is circumscribed to a circle. The tangent points from the second circle produce another quadrilateral $$KLMN$$ such that $$S_{ABCD}=3S_{KLMN}.$$ Also $$\gamma$$ is the angle between diagonals $$AC$$ and $$BD.$$ Find the area of $$ABCD.$$

I have no ideas. I wonder if I have to search any regularities of $$ABCD.$$ All given elements seem to me "distanced" from each other.
A quadrilateral of this kind is called bicentric. You might find some useful information at Bicentric quadrilateral - Wikipedia, the free encyclopedia.
 
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