SAT1000 - P877

Geometry Level pending

Here's the definition of harmonically bisect: Given that A1,A2,A3,A4A_1, A_2, A_3, A_4 are four distinct points on the coordinate plane, if A1A3=λA1A2\overrightarrow{A_1A_3}=\lambda \overrightarrow{A_1A_2}, A1A4=μA1A2\overrightarrow{A_1A_4}=\mu \overrightarrow{A_1A_2}, 1λ+1μ=2\dfrac{1}{\lambda}+\dfrac{1}{\mu}=2, then A3,A4A_3, A_4 harmonically bisect A1,A2A_1, A_2.

Given that C(c,0),D(d,0) (c,dR)C(c,0), D(d,0)\ (c,d \in \mathbb R) harmonically bisect A(0,0),B(1,0)A(0,0), B(1,0), which choice is true?

A. C could be the midpoint of AB.A.\ \textup{C could be the midpoint of AB.}

B. D could be the midpoint of AB.B.\ \textup{D could be the midpoint of AB.}

C. C, D could be both on segment AB.C.\ \textup{C, D could be both on segment AB.}

D. C, D can’t be both on the extension line of AB.D.\ \textup{C, D can't be both on the extension line of AB.}


Have a look at my problem set: SAT 1000 problems

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