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4.2

Part of MathLinks Contest 4th

Problems(1)

0442 triangle inside triangle 4th edition Round 4 p2

Source:

5/7/2021
We say that two triangles T1T_1 and T2T_2 are contained one in each other, and we write T1⊂T2T_1 \subset T_2, if and only if all the points of the triangle T1T_1 lie on the sides or in the interior of the triangle T2T_2. Let Δ\Delta be a triangle of area SS, and let Δ1⊂Δ\Delta_1 \subset \Delta be the largest equilateral triangle with this property, and let Δ⊂Δ2\Delta \subset \Delta_2 be the smallest equilateral triangle with this property (in terms of areas). Let S1,S2S_1, S_2 be the areas of Δ1,Δ2\Delta_1, \Delta_2 respectively. Prove that S1S2=S2S_1S_2 = S^2.
Bonus question: : Does this statement hold for quadrilaterals (and squares)?
geometry4th edition