162
Question
Statement 1: If |z1|=|z2|=|z3|,z1+z2+z3=0 and (z1),B(z2),C(z3) are the vertices of ΔABC, then one of the values of arg((z2+z3−2z1)(z3−z2)) is π2.
Statement 2: In equilateral triangle, orthocenter coincides with centroid.
Statement 1: If |z1|=|z2|=|z3|,z1+z2+z3=0 and (z1),B(z2),C(z3) are the vertices of ΔABC, then one of the values of arg((z2+z3−2z1)(z3−z2)) is π2.
Statement 2: In equilateral triangle, orthocenter coincides with centroid.
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Solution
The correct option is B Both the statements are true, but Statement 2 is not the correct explanation for Statement 1.
Since |z1|=|z2|=|z3|, circumcenter of Δ is origin
Also z1+z2+z33=0
Centroid coincide with circumcenter. Hence, Δ is equilateral.
arg(z2+z3+2z1z3−z2)=arg⎛⎜
⎜⎝2⎛⎜
⎜⎝z1+z22−z1z3−z2⎞⎟
⎟⎠⎞⎟
⎟⎠
=arg⎛⎜
⎜⎝z2+z32−z1z3−z2⎞⎟
⎟⎠
(z2+z3)/2 is the midpoint of side BC. Clearly, line joining A and midpoint of BC will be perpendicular to side BC. Thus,
arg⎛⎜
⎜⎝z2+z32−z1z3−z2⎞⎟
⎟⎠=π2
Hence, statement 2 is also true. However, it does not explain statement 1.
Since |z1|=|z2|=|z3|, circumcenter of Δ is origin
Also z1+z2+z33=0
Centroid coincide with circumcenter. Hence, Δ is equilateral.
arg(z2+z3+2z1z3−z2)=arg⎛⎜ ⎜⎝2⎛⎜ ⎜⎝z1+z22−z1z3−z2⎞⎟ ⎟⎠⎞⎟ ⎟⎠
=arg⎛⎜ ⎜⎝z2+z32−z1z3−z2⎞⎟ ⎟⎠
(z2+z3)/2 is the midpoint of side BC. Clearly, line joining A and midpoint of BC will be perpendicular to side BC. Thus,
arg⎛⎜ ⎜⎝z2+z32−z1z3−z2⎞⎟ ⎟⎠=π2
Hence, statement 2 is also true. However, it does not explain statement 1.
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