 All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars  perplexus dot info  Angle Ratio Ascertainment 2 (Posted on 2015-09-26) Let PQRS be a parallelogram and T be a point on diagonal QS such that ∠TRQ = ∠ PRS.
The circumcircle of triangle PQS intersects line PR at points P and U.
Find : ∠ PUS/∠ TUQ.

** Source: Serbia National Math Olympiad.

 No Solution Yet Submitted by K Sengupta No Rating Comments: ( Back to comment list | You must be logged in to post comments.) Solution Comment 2 of 2 | Let a line from the centre, O, of the circumcircle, through the mid-
point of chord SQ cross PQ at V. Let SV and SR cross the circle
again at W and X respectively, and let WU cross SQ at Y.

arc SP = arc WQ             (symmetry about line OV)
arc SP = arc QX  (arcs between parallel chords)
/WSQ = /WUQ = /PUS = /QSX = /PQS = a say   (angles in same seg.)
and       /PSW = /PUW = /PQW = b say  (angles in same segment)

Also      /SQR = /QSP = a + b      (Alternate angles in parallelogram)

Therefore, in triangles WQY and RQT:
/WQY = /RQT     (both equal to a + b)
WQ = RQ           (both equal to SP)
/QWY = /QRT     (/QWY = /QPU angles in same segment,
/QPU = /PRS are alternate & /PRS = /QRT)

Thus, triangles WQY and RQT are congruent, and the points T and Y
coincide.
So  /TUQ = /WUQ = a = /PUS,     giving   /PUS / /TUQ = 1

Edited on October 6, 2015, 5:08 pm
 Posted by Harry on 2015-10-06 17:07:10 Please log in:

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