Number of lines equidistant from four points on a plane

In summary, the conversation is about a mathematical problem with elementary solutions from a book. The solution is on page 41. The person trying to solve the problem is struggling with understanding how there can be 4 straight lines in one case. The other person clarifies that it is either four circles or three circles and a straight line, and explains how this can be possible.
  • #1
mahblah
21
2
Homework Statement
Four points in the plane are given, not all on the same straight line, and not all on a circle. How many straight lines and circles can be drawn which are equidistant from these points?
Relevant Equations
by distance from a point P to a circle c with center O we mean the lenght of the segment PQ, where Q is the point where the ray from O in the direction OP meets c
Hi, i'm trying to solve this problem.

It's exercise 3 on page 5 from this book:
Challenging mathematical problem with elementary solutions

The solution is on page 41:

1683625713913.png

1683626064255.png


I'm OK with the 4 circles in case 1: i can pick (inside/outside):
ABC + D,
ABD + C,
ADC + B,
BCD + A.
What i cannot understand is how there can be 4 straight lines in case 1:
if three points stand on one side of the equidistant line, these point must be collinear, and so there is only one possible straight line (i cannot re-arrange them!)

1683626234473.png


where am I wrong?

thanks
 
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  • #2
The phrasing of the answer lead to a misunderstanding. It is never four straight lines, it is either four circles or three circles and a straight line. If A, B, and C are collinear, then you can draw a straight line as you did, and three circles concentric with the circles passing through ABD, ACD, and BCD.
 
  • Like
Likes mahblah
  • #3
Oh that makes sense
Thanks!
 

1. What is the definition of "Number of lines equidistant from four points on a plane"?

The number of lines equidistant from four points on a plane refers to the number of straight lines that are the same distance from all four given points.

2. How do you determine the number of lines equidistant from four points on a plane?

The number of lines equidistant from four points on a plane can be determined by finding the number of unique combinations of two points out of the four given points. This can be calculated using the formula nCr = n! / r!(n-r)!, where n is the total number of points and r is the number of points being chosen at a time.

3. Can there be more than one line equidistant from four points on a plane?

Yes, there can be more than one line equidistant from four points on a plane. In fact, there can be multiple lines if the points are arranged in a way that allows for more than one line to be the same distance from all four points.

4. How does the number of lines equidistant from four points on a plane change if one of the points is moved?

If one of the points is moved, the number of lines equidistant from the remaining three points will decrease. This is because the moved point will no longer be included in the calculation for unique combinations, resulting in a smaller number of possible lines.

5. Can the number of lines equidistant from four points on a plane be negative?

No, the number of lines equidistant from four points on a plane cannot be negative as it represents a count of the possible lines and cannot be less than zero.

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