Recapitulation

Geometry Level 5

Three circles are enclosed in a rectangle such that each circle is tangent with one another and the rectangle. The dimensions of the rectangle are $(8\sqrt6+20)$ by 24. A triangle is formed using the points of tangency of the circles as vertices.

Find the area of the triangle, which can be expressed as the expression $\frac{a\sqrt{b}}{c}$ in lowest terms, where $a$ , $b$ and $c$ are integers. Submit your answer as $a+b+c$ .

The answer is 13477.

1 solution

Maria Kozlowska
Aug 28, 2015

The radii of the circles can be obtained: $r_{1}=24/2=12, r_{2}=8, r_{3}=4.5$ The triangle in question is in fact contact triangle for the reference triangle having circle centres as its vertices. Using formula for the contact triangle area where s denotes reference triangle semiperimeter and $\triangle$ its area:

$\frac{2 \triangle ^3}{abc *s}=\frac{12096\sqrt{6} }{1375 }$

$\bigtriangleup$ area of contact triangle or bigger one? And similarly $s$ ?

Kishore S. Shenoy - 5 years, 9 months ago

bigger one which is reference triangle

Maria Kozlowska - 5 years, 8 months ago

Recapitulation

The answer is 13477.

1 solution

0 pending reports