**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

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*).

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).

**Challenges**

Can you *explain why* (prove that) all the concurrency theorems above are true?

**Explore More**

1) 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.

2) 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.

3) For a more challenging result involving four concurrent circles, go to Nine-point centre (anticentre or Euler centre) of a Cyclic Quadrilateral.

**Reference**

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.

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