To Prove: $ \frac{AP}{AQ} = \frac{MP}{NQ} = \frac{PR}{QS} $
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Step-by-Step Solution
Step 1: Identify Similar Triangles
From the given figure, we can observe two triangles, $\triangle APM$ and $\triangle AQN$. Both triangles are right-angled at $P$ and $Q$ respectively, because the radius drawn to the point of tangency is perpendicular to the tangent. Also, they share a common angle at $A$. Therefore, by AA similarity criterion, $\triangle APM$ is similar to $\triangle AQN.
Step 2: Apply Proportionality from Similar Triangles
Since $\triangle APM \sim \triangle AQN$, the ratio of their corresponding sides must be equal. Thus, the ratio of side $AP$ to $AQ$ is equal to the ratio of side $MP$ to $NQ.
Step 3: Consider the Lower Tangent
Similarly, considering the lower tangent from point $A$, we have $\triangle ARM$ and $\triangle ASN$. These triangles are also right-angled at $R$ and $S$ respectively, and share the common angle at $A$. Therefore, $\triangle ARM$ is similar to $\triangle ASN$ by AA similarity.
Step 4: Apply Proportionality for Lower Tangent
From the similarity of $\triangle ARM$ and $\triangle ASN$, we can write the ratio of corresponding sides as $\frac{AR}{AS} = \frac{MR}{NS}$. Since $MR = MP$ (radii of the smaller circle) and $NS = NQ$ (radii of the larger circle), we can substitute these into the equation.
Step 5: Relate Tangent Lengths
The lengths of tangents drawn from an external point to a circle are equal. Therefore, $AP = AR$ for the smaller circle and $AQ = AS$ for the larger circle.
Step 6: Combine Ratios
From step 2, we have $\frac{AP}{AQ} = \frac{MP}{NQ}$. From step 4, using $MR = MP$ and $NS = NQ$, we have $\frac{AR}{AS} = \frac{MP}{NQ}$. Since $AP = AR$ and $AQ = AS$, both ratios are equal to $\frac{MP}{NQ}.
Step 7: Final Proof
We have established $\frac{AP}{AQ} = \frac{MP}{NQ}$. Also, $MP$ is the radius of the smaller circle and $NQ$ is the radius of the larger circle. The segments $PR$ and $QS$ are the diameters of the respective circles, so $PR = 2MP$ and $QS = 2NQ$. Therefore, $\frac{PR}{QS} = \frac{2MP}{2NQ} = \frac{MP}{NQ}$. Combining these, we get the desired result: $\frac{AP}{AQ} = \frac{MP}{NQ} = \frac{PR}{QS}.