Christmas Streak 08/88: Six Lines + Two Circles

As you can see on the right, using the power theorem , we see that

$x(15-y)=3\cdot9 \\\\ y(15-x)=4\cdot12$

Then, it is not so hard to see that

$\begin{aligned} y(15-x)-x(15-y) &=48-27 \\\\ 15y-xy-15x+xy &=21 \\\\ 15(y-x) &=21 \\\\ y-x &= 1.4 \end{aligned}$

Finally, the value of $\color{#D61F06}\text{?}$ is

$\begin{aligned} & x+(15-y) \\\\ & = 15-(y-x) \\\\ &= 15 - 1.4 \\\\ &=\boxed{13.6}. \end{aligned}$

My approach was, since the two triangles are congruent and symmetrical, I just focus on the one circle.

By power theorem,

4(12) = (15 - (a+b))(15 - b) (1)

and

3(9) = (b)(a + b) (2).

The $\color{#D61F06}{\text{?}}$ is also the same as the value of a + 2b.

Expanding 1 yields: 48 = 225 - 15(a + 2b) + b(a + b). From 2, we can simplify it further to:

48 = 225 + 27 - 15(a + 2b).

$\begin{aligned}\therefore a + 2b &= \color{#D61F06}{\text{?}} \\ &= \dfrac{225 + 27 - 48}{15} = \boxed{13.6} . \end{aligned}$