# How to obtain non-trivial solution of equal ratios

How can I obtain a non-trivial solution of the following:

$$\frac{a}{b+c}=\frac{b}{c+a}=\frac{c}{a+b}$$

The answer is $a=b=c=1$.

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Are you sure that this is about the software Mathematica? – Öskå Aug 14 '14 at 10:12
Why $a=b=c=\frac{1}{2}$ is better than e.g. $a=b=c=1$ or just $a=b=c=d$ for any complex $d$??? – Artes Aug 14 '14 at 10:14
yes, how to program this in mathematica such that it gives the answer given. Otherwise it is giving trivial solution a = b = c = 0, but i dont know how to program for non-trivial... – xclassmechluv Aug 14 '14 at 10:14
@Artes good catch. i m sorry – xclassmechluv Aug 14 '14 at 10:16
Try e.g. FindInstance[ a/(b + c) == b/(c + a) == c/(a + b), {a, b, c}, Rationals, 5], there are infinitely many solutions. – Artes Aug 14 '14 at 10:22

Perhaps what you looking for is Reduce. It will give the whole solution space of your equations.

Reduce[a/(b + c) == b/(a + c) == c/(a + b), {a, b, c}, Reals]


This should make it clear that a == b == c == 1 is not the answer.

At the request of the OP in a comment below, here is a non-trivial solution where the three variables are not equal.

ratios = a/(b + c) == b/(a + c) == c/(a + b);
ratios /. {a -> 1, b -> -5, c -> 4}

True


The values for a, b, and c come from solution sub-space

 a > 0 && b < -a && c == (-a^2 + b^2)/(a - b)


which can be read off from the result returned by Reduce.

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I'm curious why a == b == c == 1 is not answer??? – Artes Aug 14 '14 at 10:37
@Artes. Of course it's an answer. "... not the answer" is an English idiom for "not the only answer". It is a short form of "it's an answer, but not the answer". – m_goldberg Aug 14 '14 at 10:41
@m_goldberg please suggest other answers – xclassmechluv Aug 14 '14 at 10:56
can anybody graphically present the solution set? – xclassmechluv Aug 14 '14 at 12:52
@xclassmechluv. Please stop asking extensions to your question in comments. Either formulate a new question referring to this one, or edit your question so it includes the extensions. You chances for getting the additional information you seek will be much better. – m_goldberg Aug 14 '14 at 12:56

As there are infinite solutions you must limit the scope of your search. You can find your desired a == b == c ==1 with f.e.:

 FindInstance[
a/(b + c) == b/(c + a) == c/(a + b) &&
0 < a < 10 && 0 < b < 10 && 0 < c < 10,
{a, b, c}, Integers, 9]


{{a -> 1, b -> 1, c -> 1}, {a -> 2, b -> 2, c -> 2}, {a -> 3, b -> 3, c -> 3}, {a -> 4, b -> 4, c -> 4}, {a -> 5, b -> 5, c -> 5}, {a -> 6, b -> 6, c -> 6}, {a -> 7, b -> 7, c -> 7}, {a -> 8, b -> 8, c -> 8}, {a -> 9, b -> 9, c -> 9}}

Solve[a/(b + c) == b/(c + a) == c/(a + b) &&
0 < a < 10 && 0 < b < 10 && 0 < c < 10, {a, b, c}, Integers]


Same output as above

Reduce[a/(b + c) == b/(c + a) == c/(a + b) &&
0 < a < 10 && 0 < b < 10 && 0 < c < 10, {a, b, c}]


0 < a < 10 && b == a && c == b

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Mathematica can solve this problem completely. It provides three types of non-trivial solutions.

Here we go

The two defining eqations are

eq1 = a/(b + c) == b/(c + a);

eq2 = b/(c + a) == c/(a + b);


Solving these, one by one, using Solve without specifying the variables to solve for

Solve [eq1]

{{a -> b}, {a -> -b - c}}

Solve [eq2]

{{a -> -b - c}, {b -> c}}


Solving both simultaneously, as required, gives

sol = Solve[eq1 && eq2]

{{a -> -b - c}, {a -> -(c/2), b -> -(c/2)}, {a -> c, b -> c}, {b -> -(a/2),
c -> -(a/2)}}


Check the solutions

eq1 /. sol // Simplify

{True, True, True, True}

eq2 /. sol // Simplify

{True, True, True, True}

eq1 && eq2 /. sol // Simplify

{True, True, True, True}


Now let' s look at the solutions one by one

{a, b, c} /. sol[[1]]

{-b - c, b, c}

{a, b, c} /. sol[[2]]

{-(c/2), -(c/2), c}

{a, b, c} /. sol[[3]]

{c, c, c}

{a, b, c} /. sol[[4]]

{a, -(a/2), -(a/2)}


We see that the first three solutions are structurally different from each other, the fourth is similar to solution 2.

Best regards, Wolfgang

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@ Öskå: Thanks for the editing. How did you treat the Out-parts (now in a separate Grey field indented a bit more than the command, and a thin read vertical line)? – Dr. Wolfgang Hintze Aug 14 '14 at 21:56