If perpendiculars from the ends of a diameter of any circle are drawn to a tangent of that circle, prove that the sum of the lengths of the perpendiculars is equal to the length of the diameter.

The angle formed by two tangents drawn to a circle from the same external point measures \(80^{\circ}\). Find the measure of the minor intercepted arc.

Two secant lines of the same circle share an endpoint in the exterior of the circle. Show that the product of the lengths of one secant segment and its external segment equal the product of the lengths of the other secant segment and its external segment.

If \(\mathrm{P}\) and \(\mathrm{Q}\) are points on the circumference of two concentric circles, prove that the angle included between the tangents at \(\mathrm{P}\) and \(\mathrm{Q}\) is congruent to that subtended at the center by \(\underline{P Q}\).

\(\underline{P B}\) and \(\underline{P D}\), which are secants drawn to circle \(O\), intersect the circle in points \(A\) and \(C\), respectively. In the figure shown, if \(\mathrm{PA}=4, \mathrm{AB}=5\), and \(\mathrm{PD}=12\), find \(\mathrm{PC}\)

Given: \odotP is externally tangent to \(\odot \mathrm{Q}\) at T; DTC (that is, \(\mathrm{D}, \mathrm{T}\), and \(\mathrm{C}\) are collinear \() ; \mathrm{D}\) is a point of \(\odot \mathrm{P} ; \mathrm{C}\) is a point of QQ; \(\mathrm{T}\) is the point of intersection of \(\odot \mathrm{Q}\) and $\odot \mathrm{P} . \underline{\mathrm{ACE}}\( is tangent to \)\mathrm{OQ}\( at \)\mathrm{C} ; \underline{\mathrm{BDF}}\( is tangent \)\odot \mathrm{P}\( at \)\mathrm{D} .$ Let points \(\mathrm{A}\) and \(\mathrm{B}\) be the intersection of the tangents with the common internal tangent of \(\odot \mathrm{P}\) and \(\odot \mathrm{Q}\). Prove: \(\underline{A C E} \| \underline{B D F}\)