Equality in a triangle
Prove that in any triangle it holds that:
Planes and sphere
Consider two parallel planes that have a distance and intersect a sphere of radius
. (both planes intersect the sphere.) Prove that the area of the surface of the sphere enclosed by both planes is only dependant by
and not by the position of the two planes with respect to the sphere.
Solution
WLOG we assume that the two planes which intersect the sphere of radius are perpedicular to
-axis with equations
and
respectively.
(In the scheme the points and
are points of intersection of a meridian cycle with the planes
and
respectively and the points
and
are the projections of the points
and
respectively, on
-axis.)
The area in question is the area of the surface by revolution of the meridian’s arc which lies between the planes, rotating it about the
-axis. The arc
is the graph of the function
, and the area in question is
Factorization
Let be a right triangle at
. Factor
.
Solution
The following equations hold:
(1)
(2)
(3)
We have successively:
Factorial series
Prove that
Solution
We have successively:
The composition is a metric
Let be a metric ,
be a strictly increasing function and concave on
such that
. Prove that
is a metric.