perplexus.info :: Geometry : Quad Angle

Quad Angle (Posted on 2005-12-17)

Quadrilateral ABCD is inscribed in a circle. Diagonals AC and BD meet at point Q.Rays CB and DA meet at point P. If |CD| = |CP| = |DQ|, what is the measure of angle CAD?

2001 British Mathematical Olympiad, Round 1, Problem 2.

Submitted by Bractals

Rating: 3.0000 (1 votes)

Solution: (Hide)

Since ABCD is a cyclic quadrilateral, let p = <ACB = <ADB q = <BDC = <BAC r = <CAD = <CBD s = <DBA = <DCA <CQD = s (CD = DQ) s = p + r (Ext. angle of triangle AQD) <AQB = s (Vertical angles) 2s + q = 180 (Int. angles of triangle AQB) <APB = p + q (CD = CP) r = 2p + q (Ext. angle of triangle ACP) = 2(s - r) + q 3r = 2s + q = 180 r = 60

Comments: ( You must be logged in to post comments.)

Subject	Author	Date
re(2): Solution? Spoiler?	Steve Herman	2005-12-17 23:41:34
re: Solution? Spoiler?	Charlie	2005-12-17 19:05:33
Solution? Spoiler?	Steve Herman	2005-12-17 17:00:05

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