Two circles intersect at the points $C$ and $D$. The straight lines $CD$ and $BYXA$ intersect at the point $Z$. Moreever, the straight line $WB$ is tangent to both of the circles. Suppose $ZX=ZY$ and $AB\cdot AX=100$. Find the value of $BW$.
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Tags: geometry, power of a point, radical axis
Element118
29.06.2014 16:44
SMOJ wrote: Two circles intersect at the points $C$ and $D$. The straight lines $CD$ and $BYXA$ intersect at the point $Z$. Moreever, the straight line $WB$ is tangent to both of the circles. Suppose $ZX=ZY$ and $AB\cdot AX=100$. Find the value of $BW$. Improving this question: Two circles $\Gamma_1, \Gamma_2$ intersect at the points $C$ and $D$. A line intersects $\Gamma_1$ at $A, Y$, segment $CD$ at $Z$, and $\Gamma_2$ at $X, B$, in the order $A, X, Z, Y, B$. Let $BW$ be a common tangent to both circles, with $W$ on $\Gamma_1$. Suppose $ZX=ZY$ and $AB\cdot AX=100$. Find the value of $BW$.
Dukejukem
30.06.2014 04:45
Since $Z$ lies on the radical axis of $\Gamma_1$, $\Gamma_2$, the power of $Z$ with respect to both circles is the same. Hence \[ZB \cdot ZX = ZA \cdot ZY\]\[\implies ZB = ZA.\] Hence \[100 = AB \cdot AX\]\[= AB \cdot (ZA - ZX)\]\[= AB \cdot (ZB - ZY)\]\[= AB \cdot BY.\] We recognize this as the power of $B$ with respect to $\Gamma_1.$ By Power of a Point, \[100 = AB \cdot BY = BW^2\]\[\implies BW = 10.\]