MathDB

Two straight lines are given on a plane, intersecting at point

O

at an angle

a

. Let

A

B

and

C

be three points on one of the lines, located on one side of

O

and following in the indicated order,

M

be an arbitrary point on another line, different from

O

, Let

\angle AMB=\gamma

\angle BMC = \phi

. Consider the function

F(M) = ctg \gamma + ctg \phi

. Prove that

F(M)

takes the smallest value on each of the rays into which

O

divides the second straight line. (Each has its own.) Let us denote one of these smallest values by

q

, and the other by

p

. Prove that the exprseeion

\frac{p}{q}

is independent of choice of points

A

B

and

C

. Express this relationship in terms of

a

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