3. If we substitute in (α), (β), (γ), (δ) squares for triangles, and pentagons for triangles, we have the problems for squares and pentagons respectively.
4. Every Proposition in the fourth Book is a problem.
PROP. I.—Problem.
In a given circle (ABC) to place a chord equal to a given line (D) not greater than the diameter.
Sol.—Draw any diameter AC of the circle; then, if AC be equal to D, the thing required is done; if not, from AC cut off the part AE equal to D [I. iii.]; and with A as centre and AE as radius, describe the circle EBF, cutting the circle ABC in the points B, F. Join AB. Then AB is the chord required.
Dem.—Because A is the centre of the circle EBF, AB is equal to AE [I. Def. xxxii.]; but AE is equal to D (const.); therefore AB is equal to D.
PROP. II.—Problem.
In a given circle (ABC) to inscribe a triangle equiangular to a given triangle (DEF).
Sol.—Take any point A in the circumference, and at it draw the tangent GH; then make the angle HAC equal to E, and GAB equal to F [I. xxiii.] Join BC. ABC is a triangle fulfilling the required conditions.
Dem.—The angle E is equal to HAC (const.), and HAC is equal to the angle ABC in the alternate segment [III. xxxii.]. Hence the angle E is equal to ABC. In like manner the angle F is equal to ACB. Therefore [I. xxxii.] the remaining angle D is equal to BAC. Hence the triangle ABC inscribed in the given circle is equiangular to DEF.