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Is a circumcircle real?

Home Forums Math Olympiad, I.S.I., C.M.I. Entrance Geometry Is a circumcircle real?

This topic contains 1 reply, has 2 voices, and was last updated by  swastik pramanik 8 months, 2 weeks ago.

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• #24696

Show that in any triangle, a circumcircle exists.

#24709

swastik pramanik
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This is just proving that:

Theorem:

Given any three non-collinear points $$A, B, C$$  there exists a unique circle passing through $$A, B, C$$.

Proof:

Let us consider three vertices of $$\Delta ABC$$ i.e. $$A, B, C$$. Suppose the perpendicular bisectors of $$BC$$ and $$CA$$ meet at $$S$$. Then $$S$$ lies on the perpendicular bisector of $$BC$$ implies that $$SB=SC$$. Again, $$S$$ is also a perpendicular bisector of $$CA$$  which implies that $$SC=SA$$. Hence, we have $$SA=SB=SC$$. Further more, any point equidistant from $$A, B, C$$  should lie on the perpendicular bisectors of $$BC, CA$$. Therefore, $$S$$ is the only point equidistant from $$A, B, C$$ and so the circle with centre $$S$$ and radius $$SA$$ is the unique circle passing through $$A, B, C$$.

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