Hmmm. Take a look at the example in the chart above in reply #21. You may not lose or change the shape of the contact area as measured around the circumference of the tire but you lose 2.3% in width and that smacks of changing the overall size of the contact patch.
I will admit my initial response should have been "not necessarily" than with just a "nope". Even though the contact patch size will remain the same the braking distance may be different because of tread pattern, tire material, tire structure, and how it deforms under hard braking, and of course road conditions. In my opinion, under normal road conditions, I doubt there would be much difference in braking distance.
Just to be clear, the contact patch is the area (length x width) in contact with the road at a point in time. Patch size can vary by adjusting tire pressure and via the dynamic forces under motion. For instance, look at the difference in the shape of the contact patch of a top fuel dragster's rear slicks at rest and at initial acceleration and then as its RPMs increase (
The Physics of Dragster Tires, Explained). Something similar happens with over-inflated automobile tires (and under-inflated wide tires) explaining the uneven wear along the tire's center. There are numerous papers/articles regarding this subject which are easy to find on the web and a huge topic in motorsports.
So, the example you point out is wider, but the narrower tire will have a longer (along the tread line), but narrower contact patch, while the wider will have a wider contact patch. Two like tires, same diameter, but different widths/aspect ratios, at the same tire pressure and load, will have very similarly sized contact patches even if one is 2.3% wider. In the example above, since they are not the same diameter, the total contact patch size will be different between the two tires (assuming the same load and pressure on the two tires).
