P, Q and R are three points located on a circle L with diameter 4 and satisfying PQ = QR. Point S is located inside L in such a manner that QR = RS = SQ. The line passing through P and S intersects L at the point T.

Determine the length of ST.

QRS is an equilateral triangle,

  QRS = RSQ = SQR = 60

Therefore,

  PQS = PQR - SQR = (OQP + OQR) - SQR = 2x - 60

PQS is an isosceles triangle. Therefore,

  QPS = QSP = 90 - PQS/2 = 120 - x

Therefore,

  RST = 180 - PSQ - QSR = x

PQRT is a cyclic quadrilateral. Therefore,

  QRT = 180 - QPT = 180 - QPS = 60 + x

Therefore,

  SRT = QRT - QRS = x

Therefore,

  SRT = RST implies ST = RT

Therefore,

  ORT = QRT - QRO = 60

The base angle of the isosceles triangle ROT is 60.
Therefore, ROT is an equilateral triangle with

  RT = OT

Hence,

  ST = RT = OT

ST equals the radius of circle L.

Posted by Bractals on 2008-01-29 17:30:11