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Mathematical model of whether the chair can lie flat on uneven ground
Remember that the four vertices of the quadrilateral are ABCD, the diagonal intersection is O, the stool is placed on the ground, at least three feet touch the ground, O is the rotation axis, and the initial position of AC is the polar axis. When AC turns to θ, remember that the sum of the distances from A and C to the ground is F (θ), and the sum of the distances from BD to the ground is g(θ). Because there are three feet in contact with the ground at any position, there is always F (θ) *. Obviously, F(θ) is continuous. For the initial position, let F (0) = 0 and G (0) ≥ 0, then F(0)=-g(0). When the stool turns from point D to point A, we can know from symmetry that g(θ)=f(0)=0, so f(θ)≥0.

So F(θ)*F(0)=-g(0)*f(θ)≤0. According to the intermediate value theorem of continuous function, at least one point on [0, θ] makes F(x)=0, that is, f(x)=g(x)=0, so a rectangular stool can always be placed stably.