Tangent Circ|es

Geometry Level pending

Points $A, B, C$ , and $D$ lie on a line $l$ in that order, with $AB=CD=4$ and $BC=8$ . Circles $Ω_{1}, Ω_{2}$ , and $Ω_{3}$ with diameters $AB, BC$ , and $CD$ , respectively, are drawn. A line through $A$ and tangent to $Ω_{3}$ intersects $Ω_{2}$ at the two points $X$ and $Y$ . If the length of segment $XY$ can be expressed as $\frac{a}{b}\sqrt{c}$ , where $c$ is square-free and $a$ and $b$ are coprime, find $a+b+c$ ..

The answer is 36.

1 solution

Hongqi Wang
Mar 2, 2021

line center $O_2$ of $\Omega_2$ and mid point $P$ of $XY$ , and center $O_3$ of $\Omega_3$ and tangent point $Q$ , then: $\\ \dfrac {PO_2}{QO_3} = \dfrac {AO_2}{AO_3} \\ PO_2 = \dfrac {AO_2}{AO_3} QO_3 \\ = \dfrac {4+4}{4+8+2} \times 2 = \dfrac 87 \\ \dfrac12XY = \sqrt{R_{O_2}^2 - PO_2^2} \\ = \sqrt {4^2 - (\dfrac 87)^2} \\ XY = \dfrac {24}{7}\sqrt 5$

Tangent Circ|es

The answer is 36.

1 solution

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