Geometry: Secants and Inequalities

Question

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. If S is not the centre of the circumcircle, then which of the following is true?

Accepted Answer

D. By secant property, $PS \cdot ST = QS \cdot SR$. Since $HM < GM$, $\frac{2}{\frac{1}{PS} + \frac{1}{ST}} < \sqrt{PS \cdot ST} = \sqrt{QS \cdot SR}$. Also $GM < AM \implies \sqrt{QS \cdot SR} < \frac{QS + SR}{2} = \frac{QR}{2}$. Hence $\frac{1}{PS} + \frac{1}{ST} > \frac{4}{QR}$.

Answer

\frac{1}{PS} + \frac{1}{ST} < \frac{2}{\sqrt{QS \cdot QR}}

Answer

\frac{1}{PS} + \frac{1}{ST} < \frac{4}{QR}

Answer

\frac{1}{PS} + \frac{1}{ST} > \frac{1}{\sqrt{QS \cdot QR}}

Answer

\frac{1}{PS} + \frac{1}{ST} > \frac{4}{QR}

Answer

None of the above

Geometry: Secants and Inequalities

Explanation

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