Prove that if a , b , c ∈ R are all distinct, then a + b + c = 0 if and only if ( a , a 3 ) , ( b , b 3 ) , ( c , c 3 ) are collinear.

Question

Prove that if   a  ,  b  ,  c  ∈      R   are all distinct, then   a  +  b  +  c  =  0 if and only if   (  a  ,      a    3    )  ,  (  b  ,      b    3    )  ,  (  c  ,      c    3    ) are collinear.

Ariella Bruce · Accepted Answer

Step 1The first part of your proof is enough because you can proceed by equivalence  aligned points       ⟺       equal slopes       ⟺           c    2    +  b  c  +      b    2    =      b    2    +  a  b  +      a    2         ⟺       (  a  −  c  )  (  a  +  b  +  c  )  =  0       ⟺       (  a  +  b  +  c  )  =  0Here is an alternate proof:Using the classical alignment criteria for 3 points   (      x    k    ,      y    k    ) for k=1,2,3 which is      |                                        x            1                                                x            2                                                x            3                                                            y            1                                                y            2                                                y            3                                                1                          1                          1                      |    =  0This means that we have to show that      |                            a                          b                          c                                                  a            3                                                b            3                                                c            3                                                1                          1                          1                      |    =  0       ⟺       a  +  b  +  c  =  0But this is very easy because the determinant can be factorized in the following way:  (  a  −  c  )  (  b  −  a  )  (  b  −  c  )  (  a  +  b  +  c  )  ,knowing that a,b,c are all different.In fact, I just realized that I had already answered a similar question here... with two proofs, this one and another one based on a third degree equation with no term in       x    2

Prove that if a , b , c ∈ <mrow class="MJX-TeXAtom-ORD">

Answered question

Answer & Explanation

New Questions in High school geometry