- Trending Categories
- Data Structure
- Networking
- RDBMS
- Operating System
- Java
- iOS
- HTML
- CSS
- Android
- Python
- C Programming
- C++
- C#
- MongoDB
- MySQL
- Javascript
- PHP

- Selected Reading
- UPSC IAS Exams Notes
- Developer's Best Practices
- Questions and Answers
- Effective Resume Writing
- HR Interview Questions
- Computer Glossary
- Who is Who

# If two circles intersect at two points, prove that their centres lie on the perpendicular bisector of the common chord.

Given:

Two circles intersect at two points

To do:

We have to prove that their centres lie on the perpendicular bisector of the common chord.

Solution:

Let two circles with centres $O$ and $O\'$ intersect each other at $A$ and $B$.

$OA = OB$ (Radii of the circle)

$O’A = O\'B$ (Radii of the circle)

$OO’ = OO’$ (Common side)

Therefore, by SSS congruency,

$\triangle AOO’$ and $\triangle BOO’$ are similar.

This implies,

$\triangle AOO’ \cong \triangle BOO’$

$\angle AOO’ = \angle BOO’$............(i)

In $\triangle AOC$ and \triangle BOC$,

$OA = OB$ (Radii)

$\angle AOC = \angle BOC$ ($\angle AOO’ = \angle BOO’$)

$OC = OC$ (Common side)

Therefore, by SAS congruency,

$\triangle AOC \cong \triangle BOC$

This implies,

$\angle ACO = \angle BCO$

$\angle ACO+\angle BCO = 180^o$

$2\angle ACO=180^o$

$\angle ACO = \angle BCO = \frac{180^o}{2} = 90^o$

$OO’$ is the perpendicular bisector of $AB$.

Hence, their centres lie on the perpendicular bisector of the common chord.

- Related Questions & Answers
- Check if two line segments intersect
- Program to count number of points that lie on a line in Python
- Intersect two lists in C#
- Checking if decimals share at least two common 1 bits in JavaScript
- Print the kth common factor of two numbers
- Python - Intersect two dictionaries through keys
- C# program to check if two lists have at-least one element common
- Python program to check if two lists have at least one common element
- C++ Program for the Common Divisors of Two Numbers?
- Queries on count of points lie inside a circle in C++
- Program to calculate the area between two Concentric Circles in C++?
- Prime points (Points that split a number into two primes) in C++
- Finding two numbers given their sum and Highest Common Factor using JavaScript
- Check if a given circle lies completely inside the ring formed by two concentric circles in C++
- Number of Integral Points between Two Points in C++
- Maximum points of intersection n circles in C++