Mickey Mouse?

Geometry Level 3

If $a,b,$ and $c$ are the side lengths of $\triangle ABC$ opposite to angles $A,B,$ and $C,$ respectively, and $r_{1},r_{2},$ and $r_{3}$ are the corresponding exradii, then find the value of

$\frac {b-c}{r_{1}} + \frac {c-a}{r_{2}} + \frac{a-b}{r_{3}}.$

The answer is 0.

3 solutions

Purushottam Abhisheikh
Jan 30, 2015

Since $r_{1} = \frac{\Delta}{s-a}, r_{2} = \frac{\Delta}{s-b}, r_{3} = \frac{\Delta}{s-c}$ (where $\Delta$ and $s$ are area and semiperimeter of the triangle). Put these in the question. After expanding all terms would cancel out.

Jackson Abascal
Jan 31, 2015

Since the information given in the problem implies the formula holds the same value for all triangles, we can consider the specific case of an equilateral triangle a=b=c and watch everything cancel out instantly to 0.

I also did this way initially.

Purushottam Abhisheikh - 6 years, 4 months ago

This method is good for competitive exams, with MCQ question. I will use this for multiply Olympiads. Thanks for sharing.

Prayas Rautray - 3 years, 10 months ago

Michael Davis
Feb 2, 2015

The third addend in the equation becomes zero in both an equilateral and (what I used) an isoceles. I like both the solutions below. Bravo!

Mickey Mouse?

The answer is 0.

3 solutions

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