Lean Napoleon's Triangles: What is this about?
A Mathematical Droodle
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Copyright © 19962017 Alexander BogomolnyThe applet attempts to suggest the following statement [Garfunkel, Stahl]:
On the middle thirds of ΔABC construct (similarly oriented) equilateral triangle, KLF, MND, and PQG. Then ΔGDF is equilateral.
Proof
The proof is said to be computational but not tedious.
ΔAKF is isosceles, with ∠AKF = 120°. So its base angles (KAF and AFK) are both 30°. It follows that ∠AFL = 90° which makes ΔAFL right. If, as usual,Similarly, ΔAMD is right and (AD)² = b²/3.
We are now in a position to apply the Law of Cosines in ΔADF to determine the length of DF. Note that

where S is the area of ΔABC. (We again used Law of Cosines and also a formula for the area of a triangle.) The expression is symmetric in a, b, c so that (GD)² and (GF)² are bound to be equal to the same quantity, proving that indeed ΔGDF is equilateral.
Observe that the above derivation only works when triangles KLF, MND, and PQG are drawn outwardly on the sides of ΔABC. If they are drawn inwardly, then ∠DAF may equal either

with the same conclusion. Thus we are led to two equilateral triangles related to a base ΔABC. From the above relations it follows that the difference in areas of the two triangles is exactly that of ΔABC.
As James Bond drops on an occasion in "Tomorrow Never Dies", "They'll publish anything these days."
As a matter of fact, the triangle GDF is exactly Napoleon's triangle, because points G, D, F are the centers of the equilateral triangle formed on the sides of ΔABC:
References
 J. Garfunkel, S. Stahl, The Triangle Reinvestigated, Am Math Monthly, Vol. 72, No. 1. (Jan., 1965), pp. 1220
Napoleon's Theorem
 Napoleon's Theorem
 A proof with complex numbers
 A second proof with complex numbers
 A third proof with complex numbers
 Napoleon's Theorem, Two Simple Proofs
 Napoleon's Theorem via Inscribed Angles
 A Generalization
 Douglas' Generalization
 Napoleon's Propeller
 Napoleon's Theorem by Plane Tessellation
 Fermat's point
 Kiepert's theorem
 Lean Napoleon's Triangles
 Napoleon's Theorem by Transformation
 Napoleon's Theorem via Two Rotations
 Napoleon on Hinges
 Napoleon on Hinges in GeoGebra
 Napoleon's Relatives
 NapoleonBarlotti Theorem
 Some Properties of Napoleon's Configuration
 Fermat Points and Concurrent Euler Lines I
 Fermat Points and Concurrent Euler Lines II
 Escher's Theorem
 Circle Chains on Napoleon Triangles
 Napoleon's Theorem by Vectors and Trigonometry
 An Extra Triple of Equilateral Triangles for Napoleon
 Joined Common Chords of Napoleon's Circumcircles
 Napoleon's Hexagon
 Fermat's Hexagon
 Lighthouse at Fermat Points
 Midpoint Reciprocity in Napoleon's Configuration
 Another Equilateral Triangle in Napoleon's Configuration
 Yet Another Analytic Proof of Napoleon's Theorem
 Leo Giugiuc's Proof of Napoleon's Theorem
 Gregoie Nicollier's Proof of Napoleon's Theorem
 Fermat Point Several Times Over
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Copyright © 19962017 Alexander Bogomolny62360507 