Isotomy and Isogonality Hand-in-Hand: What is this about?
A Mathematical Droodle

Explanation

The forgoing statement generalizes Kiepert's theorem which follows when KL span the whole AB, etc.

In [Garfunkel, Stahl], the concurrency is observed and proven when the named segments coincide with the middle thirds of the corresponding sides and the constructed triangles are equilateral. We prefer to treat a more general theorem.

Proof

A proof follows directly from a theorem by D. Gale.

To use the theorem, we have to show that the segments AB' and AC' are isogonal conjugate as are the pairs CB', CA' and BA', BC'.

Since K and L are isotomic, AK/AB = LB/AB, so that LB/AB = BP/BC. Also, KL/AB = PQ/BC. Since triangles KLC' and PQA' are similar, we get another proportion: C'L/AB = A'Q/BC. Angles BLC' and BPA' are equal as being supplementary to equal angles. From which we conclude that triangles BLC' and BPA' are similar, implying that angles LBC' and PBA' are equal. Thus BA' and BC' are indeed isogonal conjugates.

For the pairs CB', CA' and BA', BC' the derivation is similar and we are in a position to apply D. Gale's theorem.

References

1. J. Garfunkel, S. Stahl, The Triangle Reinvestigated, Am Math Monthly, Vol. 72, No. 1. (Jan., 1965), pp. 12-20

Napoleon's Theorem

• A proof by tessellation
• A proof with complex numbers
• A second proof with complex numbers
• Two proofs
• Inscribed angles
• Douglas' Generalization
• Napoleon's Theorem by Transformation
• A Generalization
• Napoleon's Propeller
• Fermat point
• Kiepert's theorem
• Lean Napoleon's Triangles