MathDB

Consider an arc of a planar curve; let the radius of curvature at any point of the arc be a differentiable function of the arc length and its derivative be everywhere different from zero; moreover, let the total curvature be less than

\frac{\pi}{2}

. Let

P_1,P_2,P_3,P_4,P_5

and

P_6

be any points on this arc, subject to the only condition that the radius of curvature at

P_k

is greater than at

P_j

j<k

. Prove that the radius of the circle passing through the points

P_1,P_3

and

P_5

is less than the radius of the circle through

P_2,P_4

and

P_6

Problem Statement