The results below generalize the 6-point Pellegrinetti circle for the basic Van Aubel configuration for a quadrilateral with squares constructed on the sides [2] to similar parallelograms constructed on the sides [1, 3]. See this dynamic webpage for more background information on Van Aubel's theorem for squares & other generalizations & related results.

**Theorem 1**: Given similar parallelograms with centres *G*, *H*, *I* and *L* constructed on the sides of a quadrilateral *ABCD* as shown below. If *V* is the intersection of *GI* and *HL*, and *E* and *F* are respective midpoints of *AC* and *BD*, and *W* and *U* are the respective midpoints of *GI* and *HL*, then *V*, *U*, *F*, *W*, and *E* are concyclic (red circle).

**Theorem 2**: Given the same configuration as before with *G'*, *H'*, *I'* and *L'* the respective reflections of *G*, *H*, *I* and *L* in the sides of the quadrilateral. If *V'* is the intersection of *G'I'* and *H'L'*, and *E* and *F* are respective midpoints of *AC* and *BD*, and *W'* and and *U'* are the respective midpoints of *G'I'* and *H'L'*, then *V'*, *U'*, *F*, *W'*, and *E* are concyclic (blue circle).

**Theorem 3**: Lastly, the blue and red circles lie symmetrically in respect of the Newton-Gauss line *EF*, and the radius of the two circles equals ½*EF*/sin (∠*AGX*).

Van Aubel Twin Circles

**Theorem 4**: Given the same configuration as before with *G''*, *H''*, *I''* and *L''* the respective reflections of *G*, *H*, *I* and *L* in the *perpendicular bisectors* of the sides of the quadrilateral (click the '**Show More Concyclic Points**' button). If *V''* is the intersection of *G''I''* and *H''L''*, and *W''* and *U''* are the respective midpoints of *G''I''* and *H''L''*, then *V''*, *U''*, and *W''* are concyclic with the red circle. The same result holds if the points *G'*, *H'*, *I'* and *L'* in Theorem 2 are reflected in the *perpendicular bisectors* of the sides of the quadrilateral, in which case they will lie on the blue circle (not shown).

**Theorem 5**: Given the same configuration as before with *G'''*, *H'''*, *I'''* and *L'''* the respective centres of 'exterior' parallelograms constructed on the sides of the quadrilateral, similar to the original, but with the sides in inverted ratio (click the '**Show Another Concyclic Point**' button). If *V'''* is the intersection of *G'''I'''* and *H'''L'''*, and *W'''* and *U'''* are the respective midpoints of *G'''I'''* and *H'''L'''*, then *V'''* is concyclic with the blue circle, *U'''*, and *W'''* respectively coincide with *U'* and *W'*. The same result holds for the 'interior' construction related to the centres *G'*, *H'*, *I'* and *L'* in Theorem 2, in which case the intersection of the lines through the newly obtained opposite centres will lie on the red circle (not shown).

**References**

1) De Villiers, M. (2000). Generalizing
Van Aubel using Duality. *Mathematics Magazine*, 73 (4), 303-307.

2) Pellegrinetti, D. (2019). The Six-Point Circle for the Quadrangle. *International Journal of Geometry*, 2, 5-13.

3) Silvester, J.R. (2006). Extensions of a theorem of Van Aubel. *The Mathematical Gazette*, 90 (517), 2-12.

**Published Paper**: A paper An extension of the six-point circle theorem for a generalised Van Aubel configuration by Dario Pellegrinetti & myself about the above results & their proofs has been published in *The Mathematical Gazette*, 106(567), November 2022, pp. 400-407. Should your university or institution not be able to provide access, a personal copy can be requested directly from me.

**Special Case**: The special case with similar rhombi on the sides is also interesting because then the two circles coincide into the 6-point Pellegrinetti circle. A dynamic sketch can be viewed and interacted with at the provided link.

**Some other related Van Aubel Circle Results**: View more circle results related to a Van Aubel configuration (with similar triangles rather than similar quadrilaterals) which follow from a useful Fundamental Theorem of Similarity by clicking on the provided links.

**Explore More**: Investigate more circles by considering the quadrilateral *GHIL* as well as the one formed by the midpoints of the segments connecting the outer, adjacent vertices of the similar parallelograms.

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Created by Michael de Villiers, 1 October 2020 with *WebSketchpad*; updated 10 November 2020; 5 March 2021; 24 November 2022.