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#### Andrei

##### Member
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
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.

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#### Andrei

##### Member
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
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.