AB = CD = a; BC = b; EF = EH = c; and EG = d.
[Omega] = area AEB + BEC + CED,
= ac + ½bd.
[chi] = 2a + b.
Putting these values in (1),
ac + ½bd = (a + ½b)d; and hence c = d.
[Illustration: FIG. 115.]
That is, EF, EG, EH are all equal, hence a semicircle struck from E
with radius equal to the depth of the stream will pass through F and H
and be tangential to the sides of the channel.
[Illustration: FIG. 116.]
To draw the channel, describe a semicircle on a horizontal line with
radius = depth of channel. The bottom will be a horizontal tangent of
that semicircle, and the sides tangents drawn at the required side
slopes.