Circle problem in which i wanted to find value of $PQ$

Find a value of $$PQ$$:

Let the radius of bigger circle be $$R$$, and that of the smaller circles be $$r_1$$ and $$r_2$$.

Hence we can find value of $$R = 10$$. I don't know how to proceed further to find the Value of $$r_1$$ and $$r_2$$.

• PQ is coming to be $$\sqrt{4r_1 r_2}$$. – maveric Mar 5 '19 at 11:20

Let $$AP =x$$, $$PQ = y$$ and $$QB = z$$, so by Pythagoras theorem we have $$x+y+z = 20$$

Since $$AC = AQ$$ and $$BP = BC$$ we have also: $$x+y=12$$ $$y+z=16$$

so $$PQ = y = 8$$.

I used a following lemmas:

Lemma 1: $$A,H,G$$ are collinear.

Proof: Observe a homothety at $$G$$ wich takes smaller circle to a bigger and let $$H'$$ be a image of $$H$$. Then $$H'$$ is on bigger circle and tangent $$CH$$ goes to parallel tangent on bigger circle, so it can go only through $$A$$, so $$H'=A$$ and thus $$A,H,G$$ are collinear.

Lemma 2: $$AF = AC$$

Proof: Suppose $$A,H,G$$ are collinear. Since $$BGHD$$ is cyclic we can use the power of the point $$A$$: $$AH \cdot AG = AD\cdot AB$$ If we use the power of the point $$A$$ with respect to smaller circle we have:

$$AH\cdot AG = AF^2$$

And if we use the power of the point $$A$$ with respect to circle $$(BCD)$$ which is tangent to $$AC$$ we have: $$AD\cdot AB = AC^2$$

So $$AC = AF$$.

Using the power of the point $$O$$ with respect to circle $$O_1$$ we have

\begin{align} |OT_2|\cdot|OT_1|&=|OP|^2 ,\\ \text{or }\quad R(R-2r_1)&=(R-(|AD|-r_1))^2 ,\\ r_1&= \sqrt{2R(2R-|AD|)}-(2R-|AD|) \tag{1}\label{1} . \end{align}

Similarly,

\begin{align} r_2&=\sqrt{2R\cdot|AD|}-|AD| \tag{2}\label{2} . \end{align}

Note that given $$a=16=4\cdot4$$, $$b=12=4\cdot3$$ means that $$\triangle ABC$$ is a scaled version of the famous $$3-4-5$$ triangle, so $$R$$ and $$|AD|$$ can be easily calculated,

\begin{align} R&=4\cdot\frac 52=10 ,\\ |AD|&=4\cdot3\cdot\frac 35=\frac{36}5 ,\\ r_1&=\frac{16}5 ,\\ r_2&=\frac{24}5 ,\\ |PQ|&=8 . \end{align}

• Nice +1.................. – Aqua Mar 6 '19 at 14:12

I am using the lemmas provided by @greedoid . Then the calculation of PQ can be simplified as:-

$$PQ = AQ + BP - AB = 12 + 16 - 20 = 8$$