Proving that a line is straight

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SUMMARY

This discussion focuses on proving that three points are collinear using mathematical theorems and properties. Key methods include calculating the slopes between pairs of points; if the slopes are equal, the points are on the same line. Additionally, the area of the triangle formed by the three points must equal zero, and the vector cross product of vectors joining the points should also equal zero. These methods provide definitive criteria for establishing collinearity among points in a plane.

PREREQUISITES
  • Understanding of slope calculation between two points
  • Knowledge of the area of a triangle formula
  • Familiarity with vector cross product concepts
  • Basic principles of linear equations
NEXT STEPS
  • Study the concept of collinearity in geometry
  • Learn about the properties of linear equations and their graphs
  • Explore vector mathematics, specifically the vector cross product
  • Investigate the area of triangles formed by points in a coordinate plane
USEFUL FOR

Mathematicians, geometry students, educators, and anyone interested in understanding the properties of points in a plane and their relationships.

daniel_i_l
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In various riddles and math questions you're asked to prove that 3 points are on the same line. what theorms can be used to prove this sort of thing. in other words, what properties of the points can be used to prove that they're all on a line?
Thanks.
 
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Choose two of them find the slope of those two points, pick two others(one of the originals and the other one) and find the slope between those two, if the slopes are the same they lie on the same line.
 
d_leet said:
Choose two of them find the slope of those two points, pick two others(one of the originals and the other one) and find the slope between those two, if the slopes are the same they lie on the same line.

Ah... No.

Parallel lines have the same slope, but never intersect.

Find the equation of a line between 2 of the points.

Do the other points satisfy the equation? If so, they are on the line.
 
Some other properties that might be useful:

Area of the triangle defined by the 3 points = 0

Vector cross product of two vectors joining the points = 0

If the points have position vectors a b and c in order along a line, then b = ka + (1-k)c for some constant k with 0 < k < 1.
 
Integral said:
Parallel lines have the same slope, but never intersect.

True, but in d_leet's method the two lines always intersect because they have a common point.
 

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