## Twin Circles for a Van Aubel configuration involving Similar Parallelograms

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 ½EF/sin (∠AGX).

#### .sketch_canvas { border: medium solid lightgray; display: inline-block; } 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.