# Prove that the line of centres of two intersecting circles subtends equal angles at the two points of intersection.

To do:

We have to prove that the line of centres of two intersecting circles subtends equal angles at the two points of intersection.

Solution:

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

In $\triangle BAA’$ and $\triangle B'AA’$

$AB = AB'$ (Radii of circle with centre $A$)

$A’B = A’B'$ (Radii of circle with centre $A'$)

$AA’ = AA’$ (Common side)

Therefore, by $SSS$ congruency,

$\triangle BAA’ \cong \triangle B'AA’$

This implies,

$\angle ABA' = \angle AB'A’$

From above,

The line of centres of two intersecting circles subtends equal angles at the two points of intersection.

Hence proved.

- Related Articles
- Recall that two circles are congruent if they have the same radii. Prove that equal chords of congruent circles subtend equal angles at their centres.
- If two circles intersect at two points, prove that their centres lie on the perpendicular bisector of the common chord.
- Prove that if chords of congruent circles subtend equal angles at their centres, then the chords are equal.
- Prove that if chords of congruent circles subtend equal angles at their centres, then chords are equal.
- If from any point on the common chord of two intersecting circles, tangents be drawn to the circles, prove that they are equal.
- If two equal chords of a circle intersect within the circle, prove that the line joining the point of intersection to the centre makes equal angles with the chords.
- Prove that two different circles cannot intersect each other at more than two points.
- In the figure, $O$ and $O’$ are centres of two circles intersecting at $B$ and $C$. $ACD$ is a straight line, find $x$."\n
- If circles are drawn taking two sides of a triangle as diameters, prove that the point of intersection of these circles lie on the third side.
- Prove analytically that the line segment joining the middle points of two sides of a triangle is equal to half of the third side.
- Prove that the tangents drawn at the end points of a chord of a circle make equal angles with the chord.
- 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.
- Two circles of radii \( 5 \mathrm{~cm} \) and \( 3 \mathrm{~cm} \) intersect at two points and the distance between their centres is \( 4 \mathrm{~cm} \). Find the length of the common chord.
- Maximum points of intersection n circles in C++
- Prove that if the two arms of an angle are perpendicular to the two arms of another angle, then the angles are either equal or supplementary.

##### Kickstart Your Career

Get certified by completing the course

Get Started