Through the point of intersection of the circle and the bisector of the inscribed angle, a chord is drawn parallel to the side

| August 16, 2021 | education

Through the point of intersection of the circle and the bisector of the inscribed angle, a chord is drawn parallel to the side of this angle. Prove that the drawn chord is equal to the chord of the other side of the angle.

Since the OS is parallel to BD, then the angle BOC = DBO as criss-crossing angles at the intersection of parallel straight lines OS and BD of the secant AB, then the angle DBO = AOB. Then the degree measure of the arc OD = AB.

Consider triangles ОDВ and ОАВ and prove that they are equal.

The angle ОDВ and ОАВ rest on one arc ОВ, then the angle ОDВ = ОАD.

The angle ОDВ is based on the arc DB, the degree measure of which is equal to: 360 – OCB – ОD.

The angle OВA rests on the arc OA, the degree measure of which is equal to: 360 – OCB – AB.

Since the arc DO = AB, the angle ODB = OBA, which means that the triangles ODB and OAB are equal in three angles. Then BD = OA, as required.



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