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Inscribed and circumscribed quadrilateral

Andrei

Member
Jan 18, 2013
36
I would like to discuss the following problem.

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

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

earboth

Active member
Jan 30, 2012
74
I would like to discuss the following problem.

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

I have no ideas. I wonder if I have to search any regularities of \(\displaystyle ABCD.\) 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 \(\displaystyle S_{ABCD}\) means I can't give you any calculations.
 

Attachments

Andrei

Member
Jan 18, 2013
36
1. For symmetry reasons I assumed that the quadrilateral in question must be a trapezium.
...
3. ... I don't know what \(\displaystyle S_{ABCD}\) means ...
The red quadrilateral in your picture can also be circumscribed. \(\displaystyle S\) is the area.
 

Opalg

MHB Oldtimer
Staff member
Feb 7, 2012
2,703
I would like to discuss the following problem.

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

I have no ideas. I wonder if I have to search any regularities of \(\displaystyle ABCD.\) 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.