Circle Geometry: Determine Size of Angle in Terms of Unknown

**Lukybear** · July 11th 2010, 05:29 PM

A chord BC and the diameter from A meet at a point P such that OC = CP. If angle CPO = a, prove that angle AOB = 3a. This is a method of trisection an angle by construction.

What does this trisection of angle mean? Are there any other methods of solution?

**red_dog** · July 11th 2010, 10:26 PM

Let D be the other point of intersection between AP and the circle.

We have $m(\widehat{AOB})=m(arcAB)=x$

Then:

$a=m(\widehat{BPA})=\displaystyle\frac{m(arcAB)-m(arcCD)}{2}=\displaystyle\frac{x-a}{2}$

$a=\displaystyle\frac{x-a}{2}\Rightarrow 2a=x-a\Rightarrow x=3a$

**Lukybear** · July 12th 2010, 12:47 AM

Sorry, i am just not seeing it. Mabey its the notations, that i am not use to.

Can you just tell me what arcCD represents? Is it subtended at centre or circumference.
And i can guess that m is the angle size?

And also, the 2nd line, containing m(arc AB) - m(arc CD), what rule is that? Just point me in the right direction?

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