Inscribed and circumscribed quadrilateral

[Andrei1]

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]

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.

 

[Andrei1]

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]

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.

 

[CellCoree]

I can offer some insights and suggestions for approaching this problem. First, let's define some terms to better understand the problem. An inscribed quadrilateral is a quadrilateral whose vertices all lie on a circle. A circumscribed quadrilateral is a quadrilateral whose sides are all tangent to a circle.

In this problem, we are given a quadrilateral ABCD that is both inscribed and circumscribed to circles. We are also given a second quadrilateral KLMN, formed by the tangent points from the second circle, and told that the area of ABCD is three times the area of KLMN. Additionally, we are given the angle \gamma between the diagonals AC and BD.

To find the area of ABCD, we need to first understand the relationship between the two quadrilaterals. One approach could be to use the fact that the area of a quadrilateral can be calculated by dividing it into two triangles and finding the sum of their areas. We can then use the fact that the area of a triangle is equal to half the product of its base and height.

Another approach could be to consider the properties of inscribed and circumscribed quadrilaterals. For example, we know that the opposite angles in an inscribed quadrilateral are supplementary (add up to 180 degrees). We can also use the fact that the angle between a tangent and a radius of a circle is always 90 degrees.

We can also use the given information about the angle \gamma between the diagonals AC and BD to find the relationship between the two quadrilaterals. For example, if we know that one of the quadrilaterals is a rectangle, then we can use the properties of rectangles to find the area of the other quadrilateral.

In summary, to find the area of ABCD, we can use the properties of inscribed and circumscribed quadrilaterals, as well as the given information about the relationship between the two quadrilaterals and the angle \gamma between the diagonals. By combining these approaches, we can develop a method for finding the area of ABCD and potentially uncover any regularities or patterns in the problem.