Problem 5

Construct a triangle

ABC

AC=b

AB=c

, and

\angle AMB=\omega

with

\omega<90^\circ

, where

M

is the midpoint of

BC

. Prove that the construction has a solution if and only if

b\tan(\omega/2)\le c<b

. In what case does equality hold?

Step 1 of 6: Create a symmetric auxiliary segment

In plain words

Making $A$ the midpoint of $BD$ turns the angle condition at the midpoint $M$ into a constant inscribed-angle condition.

BD=2AB=2c,\qquad A\text{ is the midpoint of }BD

Detailed analysis

Extend the line $BA$ beyond $A$ to $D$ so that $BD=2AB$ . Then $A$ is the midpoint of $BD$ , exactly as in the source construction.