Let-P-1-P-2-P-n-be-a-convex-polygon-with-the-following-property-for-any-two-vertices-P-i-and-P-j-there-exists-a-vertex-P-k-such-that-the-segment-P-i-P-j-is-seen-from-P-k-under-an-ang

Tinku Tara June 4, 2023 Geometry 0 Comments

Question Number 16875 by Tinkutara last updated on 27/Jun/17

Let P_1 , P_2 , ..., P_n be a convex polygon with the following property : for any two vertices P_i and P_j , there exists a vertex P_k such that the segment P_i P_j is seen from P_k under an angle of 60°. Prove that the polygon is an equilateral triangle.

$$\mathrm{Let}\:{P}_{\mathrm{1}} ,\:{P}_{\mathrm{2}} ,\:…,\:{P}_{{n}} \:\mathrm{be}\:\mathrm{a}\:\mathrm{convex}\:\mathrm{polygon} \\ $$$$\mathrm{with}\:\mathrm{the}\:\mathrm{following}\:\mathrm{property}\::\:\mathrm{for}\:\mathrm{any} \\ $$$$\mathrm{two}\:\mathrm{vertices}\:{P}_{{i}} \:\mathrm{and}\:{P}_{{j}} ,\:\mathrm{there}\:\mathrm{exists}\:\mathrm{a} \\ $$$$\mathrm{vertex}\:{P}_{{k}} \:\mathrm{such}\:\mathrm{that}\:\mathrm{the}\:\mathrm{segment}\:{P}_{{i}} {P}_{{j}} \\ $$$$\mathrm{is}\:\mathrm{seen}\:\mathrm{from}\:{P}_{{k}} \:\mathrm{under}\:\mathrm{an}\:\mathrm{angle}\:\mathrm{of}\:\mathrm{60}°. \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{the}\:\mathrm{polygon}\:\mathrm{is}\:\mathrm{an} \\ $$$$\mathrm{equilateral}\:\mathrm{triangle}. \\ $$

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Let-P-1-P-2-P-n-be-a-convex-polygon-with-the-following-property-for-any-two-vertices-P-i-and-P-j-there-exists-a-vertex-P-k-such-that-the-segment-P-i-P-j-is-seen-from-P-k-under-an-ang

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