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 3 of 6: Intersect with the prescribed-radius circle

Previous step Next step

C\in\odot(A,b)\cap\text{(major arc }BD\text{)},\qquad AC=b

Detailed analysis

Draw the circle centered at $A$ with radius $b$ . Choose an intersection $C$ of this circle with the major arc $BD$ , and let $M$ be the midpoint of $BC$ . In triangle $BCD$ , the points $A$ and $M$ are the midpoints of $BD$ and $BC$ , so $AM\parallel CD$ ; hence $\angle AMB=\angle DCB=\omega$ . Also $AC=b$ is automatic.