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 2 of 6: Construct the constant-angle arc

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\angle BPD=\omega\quad(P\text{ on the major arc }BD)

Detailed analysis

Construct the circle through $B,D$ whose minor arc $BD$ has measure $2\omega$ . Every point $P$ on its major arc sees the chord $BD$ under the inscribed angle $\angle BPD=\omega$ .