O lvl A Maths: Plane Geometry

Question from http://www.sgforums.com/forums/2297/topics/360786

The diagonals of cyclic quadrilateral PQRS intersect at U. The circle's tangent at R meets PS produced at T. If QR = SR, prove that

PR * ST² = UR * RT²

*************************

Answer:

angle RSQ = angle SQR (isos)
angle SQR = angle SRT (alt. segment)
Hence angle RSQ = angle SQR
==> SQ is parallel to RT
==> PSU and PTR are similar triangles

Using intercept theorem,
UR / PR = ST / PT
UR / PR = ST² / (PT)(ST)

Using tangent secant theorem,
ST * PT = RT²

Hence,
UR / PR = ST² / RT²
PR * ST² = UR * RT²

Related Articles by Categories