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| Sum the quotients, get the ratio (Posted on 2007-05-30) |
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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
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Submitted by K Sengupta
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Rating: 4.0000 (2 votes)
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Solution:
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(Hide)
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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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