1. A straight line through point P of a triangle PQR intersects the side QR at the point S and the circumcircle of the triangle PQR at the point T. lf S is not the centre of the circumcircle, then which of the following is true?
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By: anil on 05 May 2019 02.37 am
Using properties of secant, $$PS imes ST = QS imes SR$$ -------------Eqn(I) Also, for two numbers, $$PS$$ and $$ST$$, we know that harmonic mean is less than geometric mean. => $$frac{2}{frac{1}{PS} + frac{1}{ST}} < sqrt{PS imes ST}$$ => $$frac{1}{PS} + frac{1}{ST} > frac{2}{sqrt{PS imes ST}}$$ Using Eqn(I) => $$frac{1}{PS} + frac{1}{ST} > frac{2}{sqrt{QS imes SR}}$$ ------Eqn(II) Also, for two numbers, $$QS$$ and $$SR$$, geometric mean is less than arithmetic mean. => $$sqrt{QS imes SR} < frac{QS + SR}{2}$$ => $$frac{1}{sqrt{QS imes SR}} > frac{2}{QR}$$ $$(ecause QS + SR = QR)$$ Multiplying both sides by $$2$$ => $$frac{2}{sqrt{QS imes SR}} > frac{4}{QR}$$ ----------Eqn(III) From eqn(II) and (III) $$ herefore frac{1}{PS} + frac{1}{ST} > frac{4}{QR}$$
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