- 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

# Let the vertex of an angle $ \mathrm{ABC} $ be located outside a circle and let the sides of the angle intersect equal chords $ \mathrm{AD} $ and $ \mathrm{CE} $ with the circle. Prove that $ \angle \mathrm{ABC} $ is equal to half the difference of the angles subtended by the chords $ A C $ and $ D E $ at the centre.

Given:

Let the vertex of an angle \( \mathrm{ABC} \) be located outside a circle and let the sides of the angle intersect equal chords \( \mathrm{AD} \) and \( \mathrm{CE} \) with the circle.

To do:

We have to prove that \( \angle \mathrm{ABC} \) is equal to half the difference of the angles subtended by the chords \( A C \) and \( D E \) at the centre.

Solution:

$AD = CE$

We know that,

An exterior angle of a triangle is equal to the sum of interior opposite angles.

This implies, in $\triangle BAE$,

$\angle DAE = \angle ABC+\angle AEC$........(i)

$DE$ subtends $\angle DOE$ at the centre and $\angle DAE$ in the remaining part of the circle.

This implies,

$\angle DAE = \frac{1}{2}\angle DOE$.........(ii)

Similarly,

$\angle AEC = \frac{1}{2}\angle AOC$..........(iii)

From (i), (ii) and (iii), we get,

$\frac{1}{2}\angle DOE = \angle ABC+\frac{1}{2}\angle AOC$

$\angle ABC =\frac{1}{2}(\angle DOE-\angle AOC)$

Hence proved.

- Related Questions & Answers
- How to set the angle of rotation of a Circle using FabricJS?
- Find other two sides and angles of a right angle triangle in C++
- JavaScript Let
- How to set the angle of skew on the X-axis of a Circle using FabricJS?
- How to set the angle of skew on the Y-axis of a Circle using FabricJS?
- Find other two sides of a right angle triangle in C++
- Get the arc sine of an angle in Java
- Get the arc cosine of an angle in Java
- Get the Trigonometric sin of an angle in Python
- Get the Trigonometric cosine of an angle in Python
- Get the Trigonometric tangent of an angle in Python
- What are start angle and end angle of arc in HTML5 canvas?
- Difference between var and let in JavaScript
- Angle between a chord and a tangent when angle in the alternate segment is given in C++?
- Return the angle of the complex argument in Python
- Why are all the deserts located on the western sides of the continents?