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Give a geometrical construction for find...

Give a geometrical construction for finding the fourth point lyingon a circle passing through three given points, without finding the centre of the circle. Justify the construction.

Text Solution

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Let A,B,C be the given points. With B as centre and radius equal to AC draw an arc. With C as centre and AB as radius draw another arc, intersecting the previous arc at D. Then, D is the desired point.

PROOF Join BD and CD.
`Delta ABC ~=Delta DCB [ :' AB = DC , AC = DB ` and `BC = CB `]
`implies /_ BAC = /_ CDB`
Thus, BC subtends equal angles `/_ BAC` and `/_ CDB` on the same side of it.
`:. `point A,B,C,D are cyclic.
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