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Given a point A of coordinates $(a_1,b_1,c_1)$ and two points $B(x_1,y_1,z_1)$ and $C (x_2,y_2,z_2)$

What are the coordinates of $D(a_2,b_2,c_2)$ symmetry of A by the line BC?

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If $D$ is symmetry of $A$ by line $BC$, then $ABDC$ is a parallelogram.

$\vec{A}+\vec{D}=\vec{B}+\vec{C}$

From there You can manage.

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  • $\begingroup$ So a2=x1+x2-a1 and same for other coordinates? Sorry for the noob question but I haven't done geometry for a good decade so it's not so fresh in my mind! $\endgroup$ – PaddleStroke Mar 7 '20 at 16:23
  • $\begingroup$ Yes correct. Here we use parallelogram property: midpoints of their diagonal coincide. $\endgroup$ – Rezha Adrian Tanuharja Mar 7 '20 at 16:26

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