Theorem 10.12 : If the sum of a pair of opposite angles of a quadrilateral is 180º, the quadrilateral is cyclic.

Explanation

Since `A`,`B`,`C` are non-collinear, on circle passes through them. Let us draw a circle with center `O`.

Let us assume `D` does not lie on circle.

`ABCD'` is a cyclic quadrilateral.

`/_BAC+/BD'C=180`

But, given that, `/_BAC+/_BDC=180`

Thus,

`/_BD'C=/_BDC`

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