Algebra + Geometry
If a , b , and c are the side lengths of a triangle such that a 2 + b 2 + c 2 = a b + b c + c a . What type of triangle is it?
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2 solutions
I did it the same way! Thanks for sharing this interesting problem!
Great solution : )
how can you say if square of their sums is equal to zero a=b=c?
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Yes it is obvious. That is because sq of any no is positive , so all positive nos add up to zero means they are actually zero. SO I did it with this logic :)
a 2 + b 2 + c 2 = a b + b c + c a ⟹ a 2 − a b + b 2 − b c + c 2 − a c = 0 ⟹ a ( a + b ) + b ( b + c ) + c ( a + c ) = 0 ⟹ a ( a + b ) + b ( b + c ) + c ( c + a ) = 0 a + 0 b + 0 c ⟹ a + b = b + c = c + a = 0 , on relating coefficients ⟹ a = b = c ∴ It is an equilateral triangle
How can you directly compare coefficients like this on both sides?
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It is possible and allowed in maths
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So you are saying that, 1x(5^2) + 1x(5)+1 = 1x(4^2) + 3x(4) + 3 implies, "by comparing the coefficients", 1=3
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@Anand Chitrao – Your equation is literally wrong. You have 3 '1' on left side. So you literally can't decide and compare coefficients.
This solution is not quite right. It is impossible for a + b = ⋯ = 0 since a , b , c > 0 - they are sides of a triangle. You can't equate coefficients like this when there are still a s, b s and c s elsewhere in the equation.
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The answer behind question is all sides are zero. And you can equate coefficients
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If all sides were zero it would not be an equilateral triangle. See Md Zuhair's solution - it shows that this holds for any triangle side length.
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@Michael Fuller – even he stated all sides equal to zero. Look at the post and comments
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@Viki Zeta – No he didn't. The sides are not equal to zero, the squares in the equation are. Let me explain...
We must have ( a − b ) 2 ≥ 0 etc since it is squared, and since the right-hand side is 0 , then we must have a − b = a − c = b − c = 0 , NOT a = b = c = 0 .
Manipulating the correct equation, we get a = b = c , and this holds for any triangle.
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@Michael Fuller – Lol sides of triangle will always be positive. Will any positive integer add upto '0'. So the answer is all sides are zero, he mentioned that to on comments
If a 2 + b 2 + c 2 = a b + b c + c a ... Mulitplying 2 to both sides we get
2 a 2 + 2 b 2 + 2 c 2 = 2 a b + 2 b c + 2 c a . Hence we get .... => ( a − b ) 2 + ( a − c ) 2 + ( b − c ) 2 = 0
Then all are equal to zero.
Hence a = b = c . Hence it is an equilateral triangle.