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?
Explanation:
As S is not the circumcentre, PS ≠ ST and QS ≠ SR
∵ PT and QR are chords of the circle intersecting at S, PS × ST = QS × SR … (i)
We know that Arithmetic mean ≥ Geometric mean
∴PS+ST2 ≥ PS×ST
But as PS ≠ ST,
PS+ST2 > PS×ST
∴PS+ST2 > QS×SR
∴PS+ST2 > 2QS×SR
∴PS+STPS×ST > 2QS+SRQS×SR
∴1PS + 1ST > 2QS×SR ...(i)
∴ Option 1 is false.
Also,
QS+SR2 > QS×SR
∴2QR < 1QS×SR
∴4QR < 2QS×SR
∴1PS + 1ST > 2QS×SR > 4QR ...From (i)
Hence, option (d).
Note: As the result is a general one, we can, without loss of generality, consider an equilateral triangle PQR with point S being the mid-point of QR and verify all options using numbers.
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