# $A$ and $D$ are in circumference of a circle and $B$ and $C$ are its inner points such that $PA$= $12$, $\frac{AB}{CD}$ = $\frac{1}{2}$. Find $PC$

There is something misunderstanding with that question that I think it to have inadequte context or information (obviously for my little knowledge). So I couldn't solve the problem.

$$PE$$ is a tangent of the below diagram. The diameter of small circle is equal to the radius of the large circle and $$BC$$ is the diameter of the small circle. If $$PA$$ = $$12$$, $$\frac{AB}{CD}$$ = $$\frac{1}{2}$$, then what is the value of $$PC$$ ?

I denoted the center of small circle $$H$$ and frew three altitude lines $$AG$$, $$HE$$ and $$DI$$ from the vertices $$A$$ , $$H$$ and $$DI$$ and also denoted $$AB$$ = $$x$$ and $$BH$$ = $$y$$.

So, $$HE$$ = $$y$$. $$\triangle AGP$$ $$\sim$$ $$\triangle HEP$$ $$\sim$$ $$\triangle DIP$$. So, I showed the relation of $$AG$$ and $$DI$$ with $$HI$$ with the help of their proportion of length because of their similarity. But I was unable to show relation with the rest property and I couldn't find the value of $$x$$ and $$y$$.

I think I went into messed situation for solving the above problem. I don't think so even my used method is going to the right direction.

It will be very helpful for me if someone please tells me in which way should I go or where is my mistake to find the right way. Thanks in advance.

Let $$AB = x$$ and $$CD = 2x$$

You can use the power of the point with respect to both circle and point $$P$$:

$$PA\cdot PD (=PE^2)= PB\cdot PC$$ $$\implies 12(12+x+2r+2x)= (12+x)(12+x+2r)$$

so $$x+2r = 12\implies PC = 12+x+2r = 24$$

Note: Information about relation betwen both radius of the circles is irrelevant.

• Why is it true your condition? I don’t understand – Federico Fallucca Feb 14 at 16:52
• Do you know the power of the point? @FedericoFallucca – Aqua Feb 14 at 16:54
• no sorry, I don’t know this concept – Federico Fallucca Feb 14 at 16:54
• Google it or wiki – Aqua Feb 14 at 16:55
• @greedoid You are absolutely right. The relation is totally irrelevant to the problem because I made the relation according to my own view and there was no description about the relation of radius and diameter of the small and the larger circle. I will try to notice the fact and I gave the information to provide some context. Pardon me for my fault. – Anirban Niloy Feb 14 at 17:19