If d1, d2 (d2>d1 ) be …

Given: d₁, d₂ (d₂ > d₁) be the diameters of two concentric circles and C be the lengths of a chord of a circle which is tangent to the other circle.

To prove : d₂² = d₁² + c² 
Since OQ and OR are radii of two circles with diameters d$_2$  and  $d_1$  respectively,therefore,$OQ = \frac{{{d_2}}}{2},OR = \frac{{{d_1}}}{2}$ and PQ = c.
Since PQ is the tangent to the circle,
therefore, OR is perpendicular to PQ.
$\Rightarrow QR = \frac{{PQ}}{2} = \frac{c}{2}$
Using Pythagoras theorem in $\Delta$OQR, we have
OQ² = QR² + QR²
$\Rightarrow {\left( {\frac{{{d_2}}}{2}} \right)^2} = {\left( {\frac{{{d_1}}}{2}} \right)^2} + {\left( {\frac{c}{2}} \right)^2}$
$\Rightarrow \frac{1}{4}{({d_2})^2} = \frac{1}{4}{({d_1})^2} + {(c)^2}$
$\Rightarrow d_2^2 = d_1^2 + {c^2}$
Hence proved.