Plane trigonometry, by S.L. Loney.

268

TRIGONOMETRY.

[Exs. XLI.]

10. Six equal circles, each of radius a, are placed so that each
touches two others, their centres being all on the circumference of
another circle; prove that the area which they enclose is
2a2 (3 /3 - 7r).
11. From the vertex A of a triangle a straight line AD is drawn
making an angle 0 with the base and meeting it at D. Prove that the
area common to the circumscribing circles of the triangles ABD and
ACD is
4 (b2y + c2P - 2bc sin A) cosec2 0,
where B and y are the number of radians in the angles B and C respectively.
235. Dip of the Horizon.
Let 0 be a point at a distance h above the earth's
surface. Draw tangents, such as OT
and OT', to the surface of the earth..
The ends of all these tangents all J<... T.
clearly lie on a circle. This circle is
called the Offing or Visible Horizon. c
The angle that each of these tangents
OT makes with a horizontal plane POQ
is called the Dip of the Horizon.
Let r be the radius of the earth
and let B be the other end of the diameter through A.
We then have, by Euc. III. 36,
OT2 = OA. OB = h (2r + h),
so that OT= Vh (2r + h).
This gives an accurate value for OT.
In all practical cases, however, h is very small compared with r.
[r= 4000 miles nearly and h is never greater, and
generally is very considerably less, than 5 miles.]