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 4 of 6: Translate existence into a distance condition

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AX\ge AC=b>AB=c

Detailed analysis

Let $X$ be the point where the perpendicular through $A$ to the relevant chord/arc direction meets the arc. The radius- $b$ circle meets the major arc exactly when its radius is no greater than the available distance $AX$ and greater than $AB$ : $AX\ge AC=b>AB=c$ .