Let ABC be a triangle in the plane. Find circles C0;C1; : : : ;C6 such that Cj
has exactly j points in common with the boundary of ABC (this \boundary"
consists of the line segments AB, BC, CA). Is it possible to �nd a circle C7
with 7 points in common with the boundary of ABC?
(Just use trial and error for this question - draw lots of circles and triangles
how do I do this?
Of course not.
Each line can intersect any circle in at most two points.
The three lines can thus intersect a circle in at most 6 points.
how do I draw them to prove it.
To find circles C0, C1, ..., C6 that have a specific number of points in common with the boundary of the triangle ABC, you can follow these steps:
1. Start by drawing the triangle ABC on a plane. Mark the points A, B, and C.
2. Circle C0: To find a circle that has 0 points in common with the triangle's boundary, draw any circle that does not intersect or touch any of the triangle's sides.
3. Circle C1: To find a circle that has exactly 1 point in common with the boundary, draw a circle that intersects one of the sides of the triangle at a single point.
4. Circle C2: To find a circle that has exactly 2 points in common with the boundary, draw a circle that intersects two different sides of the triangle, each at a single point.
5. Circle C3: To find a circle that has exactly 3 points in common with the boundary, draw a circle that intersects three different sides of the triangle, each at a single point. Note that this can be achieved by drawing a circle that passes through one of the triangle's vertices and intersects the other two sides.
6. Repeat steps 4 and 5 to find circles C4, C5, and C6 by adjusting the position and size of the circles to meet the required criteria.
7. Circle C7: Now, to determine if it is possible to find a circle C7 with 7 points in common with the boundary of ABC, observe that the triangle's boundary consists of three line segments. Since each line segment can only intersect or touch a circle at a maximum of 2 points, it is not possible for a circle to have 7 points in common with the boundary of ABC. Therefore, a circle C7 cannot be found.
Remember, for each circle, adjust its position, size, and relationship with the triangle to satisfy the specific criteria. Trial and error can help you find suitable arrangements for the circles and the triangle.