What is the greatest number of points of intersection with three circles that have 6 intersections using four straight lines?
To find the greatest number of points of intersection with three circles that have 6 intersections using four straight lines, we can break down the problem step by step.
Step 1: Understanding the scenario
Visualize three circles intersecting with each other. Each of these circles intersects with the other two circles at two points. So, the total number of intersection points for three circles is 6.
Step 2: Finding the intersections for each circle
Let's consider each circle separately. A circle can intersect with another circle at a maximum of two points. So, the first circle can intersect with the other two circles at a maximum of four points (2 points with each circle).
Step 3: Using four straight lines
Now, with only four straight lines, we need to find ways to maximize the number of intersection points between the circles.
One approach is to create a triangle by connecting the centers of the three circles. By connecting the centers, we have three straight lines. These lines intersect at three different points inside the triangle.
Next, draw one additional line passing through two of the circles. This line intersects the two circles at two more points, resulting in a total of five intersection points.
Finally, draw the fourth line passing through the remaining circle and intersecting with the other two lines inside the triangle. This line intersects with the other lines at two more points, resulting in a total of seven intersection points.
However, since we only needed six intersection points, we can remove one of the intersection points by adjusting the positions of the circles slightly. This way, we ensure that the fourth line intersects only one of the other lines at a single point, reducing the total number of intersection points to six.
So, the greatest number of points of intersection with three circles that have 6 intersections using four straight lines is six.
Note: The approach described above is just one example to achieve the desired result. There might be other configurations or arrangements of the circles and lines that can also achieve the same outcome.