Prove that the perpendicular at the point of contact to the tangent to a circle passes through the centre.

A quadrilateral $\mathrm{}\mathrm{A}\mathrm{B}\mathrm{C}\mathrm{D}\mathrm{}$ is drawn to circumscribe a circle (see Fig.). Prove that
$\mathrm{A}\mathrm{B}+\mathrm{C}\mathrm{D}=\mathrm{A}\mathrm{D}+\mathrm{B}\mathrm{C}$
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In Fig., $\mathrm{}$$\mathrm{X}\mathrm{Y}\mathrm{}$and $\mathrm{}\mathrm{X}\mathrm{\text{'}}\mathrm{Y}\mathrm{\text{'}}$ are two parallel tangents to a circle with centre $\mathrm{}\mathrm{O}$ and another tangent $\mathrm{}\mathrm{A}\mathrm{B}$ with point of contact C intersecting $\mathrm{}\mathrm{X}\mathrm{Y}$ at $\mathrm{}\mathrm{A}$ and $\mathrm{}\mathrm{X}\mathrm{\text{'}}\mathrm{Y}\mathrm{\text{'}}$ at $\mathrm{}\mathrm{B}$. Prove that $\mathrm{}\mathrm{\angle }\mathrm{}\mathrm{A}\mathrm{O}\mathrm{B}={90}^{\mathrm{o}}$.
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A triangle $\mathrm{}\mathrm{A}\mathrm{B}\mathrm{C}$ is drawn to circumscribe a circle of radius $\mathrm{}4\mathrm{}\mathrm{c}\mathrm{m}$ such that the segments $\mathrm{}\mathrm{B}\mathrm{D}$ and $\mathrm{D}\mathrm{C}$ into which $\mathrm{}\mathrm{B}\mathrm{C}$ is divided by the point of contact $\mathrm{}\mathrm{D}$ are of lengths $\mathrm{}8\mathrm{}\mathrm{c}\mathrm{m}$ and $\mathrm{}6\mathrm{}\mathrm{c}\mathrm{m}$ respectively (see Fig.). Find the sides $\mathrm{}\mathrm{A}\mathrm{B}$ and $\mathrm{}\mathrm{A}\mathrm{C}$.