How good are you with Triangles!

Geometry Level 5

$π\left(\large\sum_{}^{a,b,c} \frac{a}{bc[\sin(C-A)\sin(A-B)(\sin B + \sin C)]}\right)$

Consider a $\triangle ABC$ . Find the above summation up to 3 decimal places. Here the summation is a cyclic summation i.e the $a$ in the first term becomes $b$ in the second and $c$ in the third. The same holds for angles.

If you think the above sum is not a constant for a given triangle then type 0.22 as your answer.

Note : $A \ne B \ne C$ or $A,B,C$ are distinct .

Feel this is a bit difficult ,try this problem

This problem is a modified one, from a book written by S.L. Loney.

The answer is 0.000.

2 solutions

Vignesh S
Apr 8, 2016

The above sum can be written as

That's my solution, please post if there are any other

Just one more thing, $A=C\neq B\implies A\neq B\neq C$ .

So, better say $|\{A, B, C\}|=3$ or that $A$ , $B$ and $C$ are distinct.

A Former Brilliant Member - 5 years, 2 months ago

I had done similar to what you have said, but I think the moderator changed it. Anyway I have stated it so that the expression remains finite. What do you mean by $| { A,B,C} | =3$ ?

Vignesh S - 5 years, 2 months ago

I meant that the cardinality of the set $\{A, B, C\}$ is three. This would imply that all the elements listed are distinct.

A Former Brilliant Member - 5 years, 2 months ago

@A Former Brilliant Member – Nice representation (for angles), and thanks for the suggestion

Vignesh S - 5 years, 2 months ago

Shourya Pandey
May 20, 2016

This is not a solution.

We exploit the fact that the given equation is (or was, if the question will be edited later) not homogenous in $a,b,c$ . So if $k$ is the answer to the question, then on doubling the sides of the triangle, $\frac {k}{2}$ is also a solution. So $k =\frac {k}{2}$ , or $k=0$ .

It is a good solution( except the fact that if its not a constant for a general triangle then you need to type 0.22 as your answer, since you have 3 chances in brilliant, you can get it right).

Vignesh S - 5 years ago

How good are you with Triangles!

The answer is 0.000.

2 solutions

0 pending reports