### Dr. Wilson

#### Prove that the deflection angle between the tangent to a circle and a chord to another point on the circle is half of the central angle determined by the chord.

This represent an extension of the fact that that the inscribed angle is half of the central angle. As one of the points on the circle gets closer and closer to the vertex of the inscribed angle, the inscribed angle gets closer and closer to the deflection angle, so it should not be surprising that the deflection angle is half of the central angle just like the inscribed angle is half of the central angle.

This fact is used by surveyors when laying out roads. With very few exceptions, all of the curves on roads are arcs of circles, and the curves are joined by straight lines that are tangent to the arcs. When surveyors are putting stakes out to mark the curves, they set up a transit at the point of tangency and use the fact that the deflection angle is half of the central angle to locate the points on the curve where they need to put their stakes.