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circles


muskan
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I really think you should practise drawing half way decent diagrams.

 

Yours are , frankly, not good enough to work from.

 

Here is a better one.

 

post-74263-0-89501600-1489271249_thumb.jpg

 

Note that I have labelled all 14 angles, even though I don't yet know if I am going to use them.

 

Now first question what is your strategy for proving that PQDR is cyclic?

 

What property of the angles of a cyclic quad do you know?

 

If you do not know the answer to this ask as it is the key to the proof.

 

Once this is answered you can assemble the necessary information.

 

There are 5 triangles, 3 quadrilaterals and 4 points where some of the angles add up to 180.

 

These will give you lots of very simple equations between some of the angles.

 

But not enough.

 

You should get used to using all the information provided in a question.

 

This analysis has not yet used the fact that P, Q and R are midpoints.

 

What do you know about lines joining midpoints of triangle sides?

 

Applying this will yield enough further equations to reduce the number of unknown angles since many can be shown to be equal with this extra information.

 

I don't see that the negative mark in post#3 is either productive or warranted so I have added a +1.

Edited by studiot
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  • 2 weeks later...

Sure.

But it's hw help.... So my proof may be rejected.

 

 

I don't think anyone will mind now, the question was so long ago that the OP cannot claim credit at school for it.

There have been 163 views so we are addressing others who may like to know.

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  • 2 months later...

You can also solve the same problem with little shorter method if You make use of circumcircle covering a right angle triangle which will be trianglr ADC and Q and mid point of AC making AQ=QC=CD hence Angle QDC=angle QCD for isosceles triangle QDC and angle RPQ=angle QCD now we will talk all in angle so ill stopt typing angle before all sets

 

RPQ + RDQ

=RPQ +RDA+ADQ

=QCD+QDA+ADR

=CDQ+QDA+ADR

=180

as all 3 angle lies on a line

 

hence we can say PQDR is cyclic

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