Question

How do you prove?

If a circle with centre O has two chords DE and FG which intersect at point H then prove that `/_DOF+/_EOG=2/_EHG`?

Answer 1

Given

A circle with centre O has two chords DE and FG which intersect at point H .

RTP
we are to prove that `/_DOF+/_EOG=2/_EHG`

Construction : D and G are joined.

Proof

Now the central `/_EOG` and peripheral `/_EDG` are on the same arc GE,

So `/_EOG=2/_EDG.....[1]`

Similarly the central `/_DOF` and peripheral `/_FGD` are on the same arc FD,

So `/_DOF=2/_FGD.....[2]`

Adding [1] and [2] we get

`/_DOF+/_EOG=2/_FGD+2/_EDG`

`=2(/_FGD+/_EDG)`

`=2(/_FGD+/_HDG)`

`=2/_EHG`
(exterior angle of a triangle is equal to sum of two of the remote interior angles of the triangle.)