- For every point P in a triangular region T, consider the shortest and longest chords of T that pass thru P. Lets define 'Chord ratio(P)' = ratio between lengths of longest and shortest chords thru P.
For any triangle T, is there a unique point in its interior that is the minimum of the chord ratio? If so, is this point a known center of T?
- Define Distance ratio(P) as the ratio between longest and shortest distances to boundary of T from P. Does this ratio too have a unique minimum point inside T? Is such a minimum a known center of T?
Note 1: Numerical experiments with general triangles strongly indicate that the above minimizing points do not coincide with the centroid, incenter or orthocenter.
Note 2: Instead of the ratios between a given pair of quantities, one can try to minimize the difference between them and that will give rise to more questions.