Napoleon's Theorem via Inscribed Angles

On each side of a triangle, erect an equilateral triangle, lying exterior to the original triangle. Then the segments connecting the circumcenters of the three equilateral triangles themselves form an equilateral triangle.

We start with the following

Theorem

Proof

Let F be the second point of intersection of the circumcircles of ΔACQ and ΔBCP. ∠BFC + ∠P = 180°. Also, ∠AFC + ∠Q = 180°. Combining these with ∠P + ∠Q + ∠R = 180° immediately yields ∠AFB + ∠R = 180°. So that F also lies on the circumcircle of ΔABR.

This simple fact has the following

Corollary 1

If the vertices A,B,C of ΔABC lie, respectively, on sides QR, RP, and PQ of ΔPQR, then the three circles PCB, CQA, and BAR have a common point.

Corollary 2

If similar triangles PCB, CQA, and BAR are erected externally on the sides of ΔABC, then the circumcircles of these three triangles have a common point. (Please note that the triangles are named in such a manner as to make it obvious that vertices P,Q, and R do not correspond to each other in the given similar triangles.)

Under the conditions of Corollary 2, let O_P, O_Q, and O_R be the circumcenters of ΔBCP, ΔACQ, and ΔABR, respectively. Consider ΔO_PO_QO_R. Since its sides are perpendicular to the common chords of the three circles, ∠O_P = ∠P, ∠O_Q = ∠Q, and ∠O_R = ∠R. Hence we obtain a proof for a generalization of Napoleon's theorem:

Corollary 3

If similar triangles PCB, CQA, and BAR are erected externally on the sides of a triangle ABC, their circumcenters form a triangle similar to the given three triangles.

Remark

Point F, the point common to the circles circumscribing triangles BCP, ACQ, and ABR is remarkable in its own right. It's known as Fermat's point and often also as the Fermat-Torricelli point. If all angles of ΔABC are less than 120°, F lies inside ΔABC and is the point, for which the sum of distances to the vertices of ΔABC is minimal.

Reference

1. H. S. M. Coxeter and S. L. Greitzer, Geometry Revisited, MAA, 1967

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
• Napoleon-Barlotti 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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