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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 sae side of the line segment; the four points are concyclic.

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Proof `:` There is one and only one circle that passes through three non-collinear points.
`:.` A circle passes through points P,S and Q. Let us assume that the fourth point R does not lie on the circle.

`:.` It lie either in the interior or in the exterior of the circle.
Let us assume that point R lies in the exterior of the circle. Let seg PR intersect the circle at point T. Draw seg QT.
`/_PSQ ~= PRQ ` ...(Given ) ...(1)
`/_ PSQ ~= /_PTQ` ....(Angles inscribed in the same arc are congruent ) ....(2)
`:. /_PQR ~= /_PTQ ` ....[From (1) and (2) ] ...(3)
`/_PTQ` is an exterior angle of `Delta TRQ `
`:. /_ PTQ gt /_ TRQ` ...(Exterior angle theorem )
i.e., `/_ PTQ gt /_ PRQ ` ...(P-T-R)
This contradicts (3)
`:.` Our assumption is false
`:.` Point P cannot lie outside the circle.
Let us assume that point R lies in the interior of the circle.
`/_PSQ ~= /_PTQ ` ....(4) ....(Angles inscribed in the same arc are congruent )
But `/_PSQ ~= /_ PRQ ` ....(Given ) ...(5)
`:.` From (4) and (5)
`/_ PTQ ~= /_PRQ ` ....(6)
`/_ PRQ ` is an exterior angle of `Delta RTQ` ,
`/_ PRQ gt /_RTQ` ...(Exterior angle theorem )
i.e., `/_ PRQ gt /_ PTQ ` ....(P-R-T)
This contradicts (6)
`:.` Point R cannot lie inside the cicle,
`:.` Point R lies on the circle.
`:.` Points P,S,R and Q are concyclic .
Note `:` The above theorem is the converse of 'Angles inscribed in the same arc are congruent '.
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