In a circle with centre `O ,` chords `A Ba n dC D` intersect inside the circumference at `Edot` Prove that `/_A O C+/_B O D=2/_A E Cdot`

Explanation

In the figure, the same arc `AC`subtends `angle AOC` on the center and `angle ABC` on the other part.
`⇒ angle AOC = 2angle ABC` [Angle subtended on the center of the circle is double the angle subtended on another part of the circle by the same arc.]
Similarly, arc `BD` subtends `angle BOD` and `angle DCB`.
`⇒ angle BOD = 2angle DCB`
On adding the above results, we get
`angle AOC + angle BOD = 2angle ABC + 2angle DCB = 2(angle ABC + angle DCB)`... (`1`)
In triangle `ECB`, by exterior angle property, we have
`angle AEC = angle ECB + angle EBC`
`⇒ angle AEC = angle DCB + angle ABC `... (`2`)
On putting the value of (`2`) in (`1`), we get
`angle AOC + angle BOD = 2angle AEC`
[Hence proved]