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We consider following constellation of four circles tangent to each other:

circles

Question: Are there some interesting relations between angles and ratios of inscribed quadrilaterals? Do they have specific "features"?

Two things are immediate: $\angle BAD = \dfrac{1}{2}(\angle DO_1A + \angle AO_2B)$ and analogously for others.
As consequence, the quadrilateral $ABCD$ has the feature to be a cyclic one, ie. its vertices lie on a circle.

But are there more interesting features of such constellation? Eg, what we know about structure quadrilateral $O_1O_2O_3O_4$? Is it a tangential quadrilateral? How to derive it from that $ABCD$ is cyclic?

Nitpick: If we specialize to constellation where additionally circle $k_1$ tangents circle $k_3$, can we then say something more?
(The "nitpick part" is a bit motivated by this question: The answer to problem adressed there is given analytically by finding intersection point of two hyperbola, but I wondering if one can argue directly analyzing angle relations of inscribed quadrilaterals.

user267839
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    Please check this link: https://www.cut-the-knot.org/Curriculum/Geometry/FourTouchingCircles2.shtml .Also read the "Relevant pages" part of the above link. Hope you find it helpful. – User Oct 13 '24 at 23:07
  • @Soheil: So my conjection was wrong. The proof presented there uses intersion transformation techniques. Do you maybe know an "elementary" proof using "angle chasing" arguments. Say, a proof of kind "assuming that we know $ABCD$ is cyclic, then $O_1O_2O_3O_4$ must be tangential quadrilateral"? What we actually need to show for latter, is that for sides we have equality $O_1O_2+ O_3O_4= O_2O_3+ O_4O_1$. – user267839 Oct 14 '24 at 08:37

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