The map below shows an east/west road connecting the towns of Acorn, Centerville, and Midland, and a north/south road from Centerville to Drake. The distances from Acorn to Centerville, from Centerville to Midland, and from Centerville to Drake are each 60 kilometers. At noon Aaron starts at Acorn and bicycles east at 17 kilometers per hour, Michael starts at Midland and bicycles west at 7 kilometers per hour, and David starts at Drake and bicycles at a constant rate in a straight line across an open field. All three bicyclists arrive at exactly the same time at a point along the road from Centerville to Midland. Find the number of kilometers that David bicycles.

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1. **Determine the meeting point of Aaron and Michael:**
   - Aaron starts at Acorn and bicycles east at 17 km/h.
   - Michael starts at Midland and bicycles west at 7 km/h.
   - The distance between Acorn and Midland is \(60 + 60 = 120\) km.
   - The combined speed of Aaron and Michael is \(17 + 7 = 24\) km/h.
   - The time it takes for them to meet is:
     \[
     \frac{120 \text{ km}}{24 \text{ km/h}} = 5 \text{ hours}
     \]

2. **Calculate the distance each cyclist travels:**
   - Aaron travels:
     \[
     17 \text{ km/h} \times 5 \text{ hours} = 85 \text{ km}
     \]
   - Michael travels:
     \[
     7 \text{ km/h} \times 5 \text{ hours} = 35 \text{ km}
     \]

3. **Determine the meeting point relative to Centerville:**
   - Since Centerville is 60 km from both Acorn and Midland, and Michael travels 35 km west from Midland, the meeting point is:
     \[
     60 \text{ km} - 35 \text{ km} = 25 \text{ km} \text{ east of Centerville}
     \]

4. **Calculate the distance David travels:**
   - David starts at Drake, which is 60 km north of Centerville.
   - The meeting point is 25 km east of Centerville.
   - Using the Pythagorean Theorem to find the straight-line distance David travels:
     \[
     \text{Distance} = \sqrt{(60 \text{ km})^2 + (25 \text{ km})^2} = \sqrt{3600 + 625} = \sqrt{4225} = 65 \text{ km}
     \]

The final answer is \(\boxed{65}\) km.