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x/a + y/b + z/c = 1 intersects the coordinate axes at A,B,C. if triangle PQR has mid points A,B,C then which of the following is/are true ;

1)ar(triangle PQR) = 2 âˆš(a²b²+b²c²+c²a²)

2) foot of normal to triangle AC from O is circumcircle of PQR

#Mathematics #Co-ordinate Geometry

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Swaraj Dalmia ·2012-03-18 10:00:36

1st one is true.
Soln:
The intercepts for plane =a,b,c
Sides of triangle ABC=(ai-bj),(bj-ck),(ck-ai)
Area of triangle ABC=1/2*|(ai-bj)*(bj-ck)|=1/2*âˆš(a²b²+b²c²+c²a²)
Area of triangle PQR=4*area of triangle ABC=2*âˆš(a²b²+b²c²+c²a²)

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