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VCE Stuff => VCE Mathematics => VCE Mathematics/Science/Technology => VCE Subjects + Help => VCE Specialist Mathematics => Topic started by: /0 on May 13, 2008, 06:14:33 pm

Title: Circle Theorems (Gen. Maths?)
Post by: /0 on May 13, 2008, 06:14:33 pm

Prove that the bisectors of the four interior angles of a quadrilateral form a cyclic quadrilateral.

I do not understand  :(

Bisectors are just lines, if it's just any quadrilateral then these lines could go anywhere?

Thanks

Title: Re: Circle Theorems (Gen. Maths?)
Post by: dcc on May 13, 2008, 06:50:34 pm

A bisector is a line, but more importantly, a bisector divides the object into two equal parts.

Title: Re: Circle Theorems (Gen. Maths?)
Post by: /0 on May 13, 2008, 08:22:48 pm

I understand how bisectors work, but in this problem the notion of bisectors creating more quadrilaterals sounds a bit loony to me.

Title: Re: Circle Theorems (Gen. Maths?)
Post by: dcc on May 13, 2008, 08:40:28 pm

Draw a quadrilateral with a line coming out of the interior angles (at half of the angle, as they bisect the angle) and draw this line out.  Doing this for all four angles, you should achieve a new quadrilateral made out of these lines.

Title: Re: Circle Theorems (Gen. Maths?)
Post by: /0 on May 13, 2008, 08:59:14 pm

That's all I get

Title: Re: Circle Theorems (Gen. Maths?)
Post by: dcc on May 13, 2008, 09:36:12 pm

There is a quadrilateral in the middle!

Title: Re: Circle Theorems (Gen. Maths?)
Post by: bigtick on May 13, 2008, 09:51:28 pm

Let the bisectors bisect the interior angles into a,a; b,b; c,c; d,d.
One of the interior angle of the centre quadrilateral is 180-(a+b), the opposite angle of the centre quad. is 180-(c+d).
Sum of the two angles = 360-(a+b+c+d)=360-180=180.
So the centre quad. is cyclic.

Title: Re: Circle Theorems (Gen. Maths?)
Post by: /0 on May 13, 2008, 10:32:16 pm

Thanks!!!