Chords AB and CD of a circle with centre O, intersect at a point E. If OE bisects angle AED, prove that chord AB = chord CD.

Answer

To prove that chord AB is equal to chord CD given that OE bisects angle AED, we can follow these steps:

### Step-by-Step Solution:

1. **Draw the Circle and Chords**:
- Draw a circle with center O.
- Draw two chords AB and CD that intersect at point E.

2. **Draw Perpendiculars from O**:
- Draw a perpendicular OM from O to chord AB, and a perpendicular ON from O to chord CD.

3. **Identify Triangles**:
- We will analyze triangles OEM and OEN.

4. **Angles in Triangles**:
- Since OM and ON are perpendicular to the chords, we have:
- ∠OME = 90°
- ∠ONE = 90°
- Therefore, ∠OME = ∠ONE.

5. **Angle Bisector**:
- Since OE bisects angle AED, we have:
- ∠AEO = ∠DEO (by the definition of angle bisector).

6. **Congruent Angles**:
- Thus, we can say:
- ∠OEM = ∠OEN (since both are equal to ∠AEO and ∠DEO respectively).

7. **Common Side**:
- The side OE is common to both triangles, so:
- OE = OE.

8. **Triangle Congruence**:
- By the Angle-Angle-Side (AAS) criterion, triangles OEM and OEN are congruent:
- ΔOEM ≅ ΔOEN.

9. **Corresponding Parts of Congruent Triangles**:
- From the congruence of triangles, we have:
- OM = ON (corresponding parts of congruent triangles).

10. **Equidistance of Chords**:
- Since OM and ON are the perpendicular distances from the center O to the chords AB and CD respectively, and OM = ON, we conclude that the chords AB and CD are equidistant from the center O.

11. **Conclusion**:
- If two chords are equidistant from the center of the circle, then they are equal in length. Therefore, we conclude:
- Chord AB = Chord CD.