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**Napoleon Converse**

If *GHI* is an equilateral triangle, and three circles with centres at *G*, *H* and *I* are drawn so that they are **concurrent** in a point, then equilateral triangles constructed on the sides of triangle *ABC* formed by the pairwise intersections of circles *I* and *G*, *G* and *H*, and *H* and *I*, respectively, are inscribed in circles *G*, *H* and *I*.

Some Napoleon Converses

**Important**: To view & manipulate the *dynamic versions* of the two converses below, navigate to them using the appropriate Link button in the **ABOVE** dynamic sketch; the pictures below are **static**.

**Napoleon General Converse**

If *GHI* is any triangle, and three circles with centres at *G*, *H* and *I* are drawn so that they are concurrent in a point, and any triangles *DBA*, *BEC* and *ACF* are drawn on the sides of triangle *ABC* formed by the pairwise intersections of circles *I* and *G*, *G* and *H*, and *H* and *I*, respectively, so that they are inscribed in circles *G*, *H* and *I*, then ∠*D* = ∠*G*, ∠*E* = ∠*H*, and ∠*F* = ∠*I*. (An obvious specialization is that if triangles *DBA*, *BEC* and *ACF* are constructed similar to *GHI*, then they are inscribed in circles *G*, *H*, and *I*).

Napoleon General Converse

Can you generalize the above result to a quadrilateral and four concurrent circles, and prove your generalization? Can you generalise further?

**Specialization: Miquel's Theorem Converse**

If *GHI* is any triangle, and three circles with centres at *G*, *H* and *I* are drawn so that they are concurrent in a point, and any point *D* on circle *G* is chosen, and line *DB* meets circle *H* again in *E*, line *EC* meets circle *I* again in *F*, then *F*, *A* and *D* are collinear, and *DEF* is similar to *GHI*.

Specialization: Miquel's Theorem Converse

Can you generalize the above Miquel Converse result to a quadrilateral and four concurrent circles, and prove it? Can you generalise further? See *"Miquel Variation"* for more information.

For several more elaborate converses than those above, read John Wetzel's (1992) AMM paper *"Converses of Napoleon's Theorem"*.

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Created by Michael de Villiers, 22 Feb 2008. Updated, 10 Aug 2015, 17 April 2020.