# On two centers of convex regions

Definition: A line segment with both end points on the boundary of a planar convex region $$C$$ is called a chord of $$C$$.

1. Consider any point $$P$$ within a given planar convex region $$C$$. From among all chords of $$C$$ that pass thru $$P$$, find that chord for which the ratio between the 2 segments into which $$P$$ divides the chord (longer seg : shorter seg) is a maximum and note this maximum. Now, how does one find and characterize the position of $$P$$ inside $$C$$ such that this maximum ratio is a minimum?

Remarks: Numerical experiments indicate that when $$C$$ is a triangle, the optimal position of $$P$$ is always the centroid of the triangle. But for general convex $$C$$, $$P$$ may not be at the center of mass.

1. Consider any $$P$$ within a given planar convex region $$C$$. For every angle of orientation measured from $$P$$, there is a unique chord of $$C$$ that passes thru $$P$$. Consider the average over the orientation of the length of the corresponding chord. Now, how does one find and characterize that position of $$P$$ such that the average length of chord thru it is maximum?

Remarks: Numerically, when $$C$$ is a triangle, the optimal position of $$P$$ does not seem to be on the centroid or incenter of $$C$$. Note also that the average length of chord thru $$P$$ can also be considered over each chord starting at each point on the boundary of $$C$$ and passing thru $$P$$ – in this case we might have a different optimal $$P$$.

Note 1: Both questions above appear to have natural higher dimensional analogs.

Note 2: (this is actually, a further question) We can also look for that interior point that minimizes the ratio between the max distance from it to the boundary of C and the min distance from it to the boundary of C. Numerically, when C is a triangle, this 'center' is not seen to coincide with centroid or incenter.

• A reasonable start would be calculating the centers for a 6-9-13 triangle, and then finding them in or adding them to the Encyclopedia of Triangle Centers: faculty.evansville.edu/ck6/encyclopedia/ETC.html Feb 8, 2021 at 13:52
• For question 1 and with C as a triangle, the desired optimal center is (numerically) coincident with centroid. For question 2, with C a 6,9,13 triangle (ie one with coordinates (0,0), (13,0), (8.2307692, 3.64066448), the optimal center is found to be approx (6.549397, 1.194187) and I could not find a triangle center coincident with this point in the 'encyclopedia'. However, am new to that resource. Feb 9, 2021 at 7:13