Some Circle Concurrency Theorems

Theorem 1
Given any △ABC with triangles ADB, BFC & CEA constructed on its sides so that ∠D + ∠E + ∠F = 360°, then the three circumcircles of ADB, BFC & CEA are concurrent (at say P).
Note that this theorem can also be formulated as follows: “Given any hexagon ADBFCE with the sum of alternate angles ∠D + ∠E + ∠F = 360°, then the three circumcircles of ADB, BFC & CEA are concurrent.” Obviously, when the hexagon ADBFCE is regular, or more generally is cyclic, the circumcircles of the triangles above (& below) will coincide.
In addition, the circumcircles of EAD, DBF & FCE are also concurrent (in a different point). Click on the ‘Show Objects’ button to also view the concurrency P' of these three circumcircles.

Some Circle Concurrency Theorems

1) Theorem 1 can be (almost) directly applied to the following two results - click on the 'Link to Fig 2' or 'Link to Fig 3' buttons in the sketch above, respectively, to navigate to each one.

Result 2
Given any △ABC with triangles AXB, BYC & CZA constructed on its sides so that ∠X + ∠Y + ∠Z = 180°, and the respective incentres D, E and F of triangles AXB, BYC and CZA are constructed, then the three circumcircles of ADB, BEC & CFA are concurrent (at say P).

Result 3
Given any △ABC with triangles AXB, BYC & CZA constructed on its sides so that ∠X + ∠Y + ∠Z = 180°, and the respective circumcentres D, E and F of triangles AXB, BYC and CZA are constructed, then the three circumcircles of ADB, BEC & CFA are concurrent (at say P).

2) Navigate to the two theorems below by respectively clicking on the 'Link to Generalization of Napoleon' or 'Link to Miquel's Theorem' buttons in the sketch above.

Generalization of Napoleon's theorem
Given any △ABC with triangles AXB, BYC & CZA constructed on its sides so that ∠X + ∠Y + ∠Z = 180°, and the respective circumcircles & circumcentres D, E and F of triangles AXB, BYC and CZA are constructed, then the three circumcircles of AXB, BYC & CZA are concurrent (at say N), and ∠FDE = ∠X, ∠DEF = ∠Y and ∠EFD = ∠Z.

Miquel's Theorem
Given any △ABC with its vertices respectively lying as shown on the sides XZ, XY & YZ of a △XYZ, and the respective circumcircles & circumcentres D, E and F of triangles AXB, BYC and CZA are constructed, then the three circumcircles of AXB, BYC & CZA are concurrent (at say N), and △DEF is similar to △XYZ. (Note that this theorem is a special case of the above).

Special Napoleon Variation
Given any △ABC with triangles ADB, BFC & CEA constructed on its sides so that ∠D = ∠E = ∠F = 120°, then the respective circumcentres D', F' and E' of the three circumcircles of ADB, BFC & CEA form an equilateral triangle.
Note that this theorem can also be formulated as follows: “Given any hexagon ADBFCE with ∠D = ∠E = ∠F = 120°, then the respective circumcentres D', F' and E' of the three circumcircles of ADB, BFC & CEA form an equilateral triangle.”
3) Click on the 'Link to Special Napoleon Variation' button to view this equilateral triangle result.
4) Click on the 'Link to Sides D'E'F'' button to display the side lengths of △D'E'F'.
5) Challenge: Can you explain why (prove that) the result is true?
6) Click on the 'Show Hint' button for a hint to prove this result.

Challenges
7) Can you explain why (prove that) all the concurrency theorems above, as well as the related similarity and equilateral triangle results, are true?

Note
The above results about the formed equilateral triangle and similar triangles can also easily be proven from Pompe's Hexagon Theorem.

Explore More
8) Consider formulating converses for, or variations of, any of the results above. See for example, Some converses of Napoleon's theorem or A variation of Miquel's theorem & its generalization.
9) Investigate other possible generalizations or general variations of the above results. See for example, Some Generalizations of Napoleon's Theorem or Related Variations & Generalizations of Napoleon's Theorem.
10) For a more challenging result involving four concurrent circles, go to Nine-point centre (anticentre or Euler centre) of a Cyclic Quadrilateral.

Published Papers
i) A paper Some Circle Concurrency Theorems with proofs of the above results has been published in the Learning & Teaching Mathematics (LTM) journal, Dec 2022, no. 33, published by the Association for Mathematics Education of South Africa.
ii) A paper "A surprise equilateral triangle" with a proof of the Special Napoleon Variation will be appearing in the Learning & Teaching Mathematics (LTM) journal, Dec 2024, no. 37, published by the Association for Mathematics Education of South Africa.

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By Michael de Villiers. Created with WebSketchpad, 26 July 2022; uodated 30 July 2022; updated 19 Jan 2023; 8 Sept 2024.