MathDB

Let there be given two lines

D_1

and

D_2

which intersect at point

O

, and a point

M

not on any of these lines. Consider two variable points

A\in D_1

and

b\in D_2

such that

M

belongs to the segment

AB

.
(a) Prove that there exists a position of

A

and

B

for which the area of triangle

OAB

is minimal. Construct such points

A

and

B

. (b) Prove that there exists a position of

A

and

B

for which the area of triangle

OAB

is minimal. Show that for such

A

and

B

, the perimeters of

\triangle OAM

and

\triangle OBM

are equal, and that

\frac{AM}{\tan\frac12\angle OAM}=\frac{BM}{\tan\frac12\angle OBM}

. Construct such points

A

and

B

Problem Statement