Hey, I'm currently struggle with these two proofs, if anyone can help me I would be very thankful.

 

1. prove that one of the roots of x^3+ax^3 +bx + c = 0 is the negative of another if and only if c=ab.

 

2. Prove that if the diagonals of a quadrilateral divide it into four triangles of equal area, then the quadrilateral is a parallelogram.

What have you done for one? how have you tried to use the hypothesis?

For #2:

 

WLOG, let ABCD be a quadrilateral. Let AC and BD intersect at E.

 

By the problem, we know that [AEB] = [bEC] = [AED] = [DEC]. Since <AEB = <DEC, we know that AE*BE = CE*ED. Similar argument gives AE*DE = CE*BE. Solving for AE in both sides, we get AE = (CE*ED)/BE and AE = (CE*BE)/ED.

 

They're equal so:

 

(CE*ED)/BE = (CE*BE)/ED --> CE*ED*ED = BE*CE*BE. Simplifying, ED^2 = BE^2 and since this is the side of the figure, we know that it has to be positive and can take square root so ED = BE. Solving for other variable will also give AE = CE. Since the diagonals bisect each other, ABCD has to be parallelogram.

 

The area ratio came from the vertical angles and the formula a*b*sinC*1/2.

for #2,

i think quadrilateral will be a square or rhombus, not a general paralellogram

for #2' date='

i think quadrilateral will be a square or rhombus, not a general paralellogram[/quote']Being a square or rhombus, makes something a paralellogram.

For number one, if the cubic has two roots a and -a, then it must have a factor of (x-a)(x+a). You should be able to use this to prove one direction. For the other direction; just make the obvious first step

Being a square or rhombus, makes something a paralellogram.

 

u can't have 4 triangles of equal areas in a general parallelogram but the question is asking to prove for general parallelogram. the question should be modified.

Hey' date=' I'm currently struggle with these two proofs, if anyone can help me I would be very thankful.

 

1. prove that one of the roots of x^3+ax^3 +bx + c = 0 is the negative of another if and only if c=ab.

 

2. Prove that if the diagonals of a quadrilateral divide it into four triangles of equal area, then the quadrilateral is a parallelogram.[/quote']

 

I just want to confirm that the exponent in the second term is a 3 just like the first term.