Posts tagged 'semicircle'

F.10 Circles and Geometric Mean

In a semicircle on diameter AB = m + n, the perpendicular altitude h = CH from a point C on the circumference to AB is the geometric mean sqrt(m*n) of the segments AH = m and HB = n, while the radius OC = (m + n)/2 is their arithmetic mean.

F.53 Geometric Half-Angle Trigonometry

In a unit semicircle of radius R = 1 and diameter AB = 2, a point P at central angle theta has altitude sin(theta), base projections 1 + cos(theta) and 1 - cos(theta), and chord |AP| = 2 cos(theta/2) = sqrt(2 + 2 cos(theta)), giving cos^2(theta/2) = (1 + cos(theta))/2 and sin^2(theta/2) = (1 - cos(theta))/2.