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Regular Tetrahedron

Source: 1997 National High School Mathematics League, Exam One, Problem 2

March 4, 2020
geometry3D geometrytetrahedron

Problem Statement

In regular tetrahedron ABCDABCD, E∈AB,F∈CDE\in AB,F\in CD, satisfying: ∣AE∣∣EB∣=∣CF∣∣FD∣=λ(λ∈R+)\frac{|AE|}{|EB|}=\frac{|CF|}{|FD|}=\lambda(\lambda\in R_+). Note that f(λ)=αλ+βλf(\lambda)=\alpha_{\lambda}+\beta_{\lambda}, where αλ=<EF,AC>,αλ=<EF,BD>\alpha_{\lambda}=<EF,AC>,\alpha_{\lambda}=<EF,BD>. (A)\text{(A)} f(λ)f(\lambda) increases in (0,+∞)(0,+\infty) (B)\text{(B)} f(λ)f(\lambda) decreases in (0,+∞)(0,+\infty) (C)\text{(C)} f(λ)f(\lambda) increases in (0,1)(0,1), decreases in (1,+∞)(1,+\infty) (D)\text{(D)} f(λ)f(\lambda) is a fixed value in (0,+∞)(0,+\infty)
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