Data Structure
Networking
RDBMS
Operating System
Java
MS Excel
iOS
HTML
CSS
Android
Python
C Programming
C++
C#
MongoDB
MySQL
Javascript
PHP
Physics
Chemistry
Biology
Mathematics
English
Economics
Psychology
Social Studies
Fashion Studies
Legal Studies
- Selected Reading
- UPSC IAS Exams Notes
- Developer's Best Practices
- Questions and Answers
- Effective Resume Writing
- HR Interview Questions
- Computer Glossary
- Who is Who
The common tangents $ A B $ and $ C D $ to two circles with centres $ O $ and $ O^{\prime} $ intersect at $ E $ between their centres. Prove that the points $ O, E $ and $ O^{\prime} $ are collinear.
Given:
The common tangents \( A B \) and \( C D \) to two circles with centres \( O \) and \( O^{\prime} \) intersect at \( E \) between their centres.
To do:
We have to prove that the points \( O, E \) and \( O^{\prime} \) are collinear.
Solution:
Join $OA$ and $OC$.
In $\triangle OAE$ and $\triangle OCE$,
$OA=OC$ (Radii of the same circle)
$OE=OE$ (Common side)
$\angle OAE=\angle OCE=90^{\circ}$
$\Rightarrow \triangle OAE \cong \triangle OCE$ (By RHS congruence)
$\Rightarrow \angle AEO=\angle CEO$ (CPCT)
Similarly,
$\angle BEO^{\prime}=\angle DEO^{\prime}$
$\angle AEC=\angle DEB$
$\Rightarrow \frac{1}{2} \angle AEC=\frac{1}{2} \angle DEB$
$\Rightarrow \angle AEO=\angle CEO=\angle BEO^{\prime}=\angle DEO^{\prime}$
Since these angles are equal and are bisected by $OE$ and $O^{\prime}E, O, E$ and $O^{\prime}$ are collinear.
25 Views
