Circle theorem?

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1 Answer
Shwetank Mauria
Jan 11, 2018

Please see below.

Explanation:

Let the arc #hat(APC)# subtend angle #/_ABC# at thecircumference and angle #/_AOC# at the center of the circle. Join #BO# and extend it to meet the circumference at #P#, as follows.
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As #AO=BO=CO# each being equal to radius, #DeltasABO# and #BCO# are isosceles triangles.

Now in #DeltaABO#, #m/_BAO=m/_ABO# and as exterior angle #/m_AOP# is equal to sum of its interior opposite angles

#/_AOP=/_BAO+/_ABO=2/_ABO# ................(1)

Similarly in #DeltaBOC#

#/_COP=/_BCO+/_CBO=2/_CBO# ................(2)

Adding (1) and (2), we get

#/_AOC=/_ABC#

#Q.E.D.#

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