Question

Two points both lie in each of two lines.

   Two points both lie in each of two lines.

Geometry

Ray C.… 1st Edition

Chapter 1, Problem 13

Instant Answer

Solved by Expert Amrita Bhasin

10/24/2019

Step 1

This is a fundamental concept in geometry. In mathematical terms, it can be written as: if $P$ and $Q$ are two distinct points, then there exists exactly one line $l$ such that both $P$ and $Q$ lie on $l$. Show more…

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Two points both lie in each of two lines.

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Key Concepts

Two Points Determine a Line

In Euclidean geometry, a fundamental concept is that exactly one unique line can be drawn through any two distinct points. This principle ensures that if two points are shared by two different lines, then the lines must be identical, since both lines would have to coincide with the unique line determined by those points.

Coincident Lines

Coincident lines refer to lines that lie exactly on top of one another, meaning they share every point in common. When two lines are found to have the same two distinct points, the uniqueness of a line through any two points implies that the lines are not separate but are instead the very same line.

Intersection of Lines

The intersection of lines deals with the points that two or more lines have in common. In this context, if two distinct points lie on both of two lines, it indicates that the lines intersect in at least those two points. However, given that two distinct points determine a unique line, this forces the conclusion that the two lines must overlap completely, reinforcing the idea of coincident lines.

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