MathDB

Given two distinct points

A,B

in the plane, for each point

C

not on the line

AB

, we denote by

G

and

I

the centroid and incenter of the triangle

ABC

, respectively.
(a) For

0<\alpha<\pi

, let

\Gamma

be the set of points

C

in the plane such that

\angle\left(\overrightarrow{CA},\overrightarrow{CB}\right)=\alpha+2k\pi

as an oriented angle, where

k\in\mathbb Z

. If

C

describes

\Gamma

, show that points

G

and

I

also descibre arcs of circles, and determine these circles. (b) Suppose that in addition

\frac\pi3<\alpha<\pi

. For which positions of

C

\Gamma

GI

minimal? (c) Let

f(\alpha)

denote the minimal

GI

from the part (b). Give

f(\alpha)

explicitly in terms of

a=AB

and

\alpha

. Find the minimum value of

f(\alpha)

for

\alpha\in\left(\frac\pi3,\pi\right)

Problem Statement