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**rptamin****Member**- Registered: 2014-07-02
- Posts: 3

Let ABCD be a cyclic quadrilateral. Let P be the intersection of $\overline{AD}$ and $\overline{BC}$, and let Q be the intersection of $\overline{AB}$ and $\overline{CD}$. Prove that the angle bisectors of $\angle DPC$ and $\angle AQD$ are perpendicular.

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 7,930

hi rptamin

Welcome to the forum.

Try this: let x be the angle at A and y be the angle at D.

Using the angle sum of a triangle, the cyclic property that opposite angles add to 180, and the bisector property you can work out all the angles in the diagram in terms of x and y. Keep working them out until you get the angle you want = 90.

You'll need a large diagram because it gets complicated writing in expressions like (x+y)/2 - 90 for one of a bisected pair.

Bob

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