### Isotomy and Isogonality Hand-in-Hand: What is this about?

A Mathematical Droodle

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Copyright © 1996-2017 Alexander BogomolnyThe applet attempts to suggest the following statement:

On each side of ΔABC, construct a pair of isotomic conjugates: K, L on AB, M, N on CA and P, Q on BC. Assume that in each case the linear barycentric coordinates of K, M, P (and hence those of L, N, Q) are the same. (Which only means that, say, |

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.

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### 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,

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

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

- 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

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