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If a line segment joining two points sub...

If a line segment joining two points subtends equal angles at two other points lying on the same side of the line segment; the four points are concyclic.

Text Solution

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Since `A`,`B`,`C` are non-collinear, one circle passes through them. raw a circle with center `O`.
Let `D` not lie on circle. Let circle intersect `AD` at `D'`.
`/_ACB=/_AD'B` (Angle in same segment)
But, `/_ACB=/_ADB`
Thus,
`/_AD'B=/_ADB`
This may only happen if and only if `D` and `D'` coincides. Thus, point `D` lie on circle.
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Theorem: 7 If the line segment joining two points subtends equal angles at two other points lying on the same side of the line segment; the four points are concyclic.i.e lie on the same circle.

Knowledge Check

  • If the line segment joining the points A(a,b) and B(c,d) subtends an angle theta at the origin, then costheta is equal to

    A
    `(ab+cd)/(sqrt((a^(2)+b^(2))(c^(2)+d^(2))))`
    B
    `(ac+bd)/(sqrt((a^(2)+b^(2))(c^(2)+d^(2))))`
    C
    `(ac-bd)/(sqrt((a^(2)+b^(2))(c^(2)+d^(2))))`
    D
    `(ad+bc)/(sqrt((a^(2)+c^(2))(b^(2)+d^(2))))`

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