A rectangle, a circle, and a triangle are drawn in a plane. what is the maximum possible number of points of intersection? The sides of the triangle are not collinear with any of the sides of the rectangle.
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To find the maximum possible number of points of intersection between the rectangle, circle, and triangle, we need to consider all possible combinations of intersection.
Let's break it down step by step:
1. Rectangle and Circle:
The maximum number of points of intersection between a rectangle and a circle is 8. This occurs when the circle is tangent to each side of the rectangle, resulting in 2 points of tangency per side, and a total of 8 points.
2. Rectangle and Triangle:
The maximum number of points of intersection between a rectangle and a triangle is 8. This is achieved when each vertex of the triangle intersects with one of the sides of the rectangle (3 points), and the remaining point of the triangle intersects with the interior of the rectangle (1 point).
3. Circle and Triangle:
The maximum number of points of intersection between a circle and a triangle is 6. This happens when each vertex of the triangle lies on the circumference of the circle (3 points), and each side of the triangle intersects with the circle in exactly one point (3 points).
Now, to find the overall maximum number of points of intersection between all three shapes, we need to consider the combinations:
- If we have 8 points of intersection between the rectangle and circle, and 8 points of intersection between the rectangle and triangle, the total number of points of intersection is at most 16.
- However, we also need to consider the intersection between the circle and the triangle. Since the maximum possible number of points of intersection between a circle and a triangle is 6, we need to subtract these 6 points from the total obtained above.
Thus, the maximum possible number of points of intersection between the rectangle, circle, and triangle is 16 - 6 = 10.