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Sum the quotients, get the ratio (Posted on 2007-05-30) Difficulty: 2 of 5
PQRS is a cyclic quadrilateral such that a, b and c are the respective perpendicular distances of P from the lines QS, QR and RS.

Show that: QS/a = QR/b + RS/c

  Submitted by K Sengupta    
Rating: 4.0000 (2 votes)
Solution: (Hide)
QS/a = cot PQS + cot PSQ

QR/b = cot PQR + cot PRQ

RS/c = cot PRS + cot PSR, with:

Angle PRQ = ANgle PSQ, and:
Angle PRS = Angle PQS

Hence:

QR/b + RS/c - QS/a
= cot PQR + cot PSR
= 0, since Angle PQR + Angle PSR = 180o

Consequently, QS/a = QR/b + RS/c

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