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tangents to sphere, minimum length

Source: France 1988 P3

May 18, 2021
geometry3D geometrysphere

Problem Statement

Consider two spheres Σ1\Sigma_1 and Σ2\Sigma_2 and a line Δ\Delta not meeting them. Let CiC_i and rir_i be the center and radius of Σi\Sigma_i, and let HiH_i and did_i be the orthogonal projection of CiC_i onto Δ\Delta and the distance of CiC_i from Δ (i=1,2)\Delta~(i=1,2). For a point MM on Δ\Delta, let δi(M)\delta_i(M) be the length of a tangent MTiMT_i to Σi\Sigma_i, where Ti∈Σi (i=1,2)T_i\in\Sigma_i~(i=1,2). Find MM on Δ\Delta for which δ1(M)+δ2(M)\delta_1(M)+\delta_2(M) is minimal.
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