A Vector Range on a Six-Pointed Star

★ 7/10plane geometry Extrema Linear Programming vectors

Problem

As shown in the figure, two congruent equilateral triangles $ABC$ and $A_1B_1C_1$ together form a six-pointed star with center $O$ ; their sides intersect at six points, which are the vertices of the inner hexagon of the star. Let $D$ be the vertex of this hexagon at which sides $AC$ and $A_1B_1$ meet.

Let $P$ be a point on the boundary of the star (the twelve-sided outline), and write

\overrightarrow{OP} = x\,\overrightarrow{OD} + y\,\overrightarrow{OC_1}.

Find the range of $x + y$ .

Answer

Solution

Difficulty	7/10
Topics	plane geometry, Extrema, Linear Programming, vectors

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