The First Six Books of the Elements of Euclid

by Euclid; John Casey

Exercises on Book IV.

1. If a circumscribed polygon be regular, the corresponding inscribed polygon is also regular, and conversely.

2. If a circumscribed triangle be isosceles, the corresponding inscribed triangle is isosceles, and conversely.

3. If the two isosceles triangles in Ex. 2 have equal vertical angles, they are both equilateral.

4. Divide an angle of an equilateral triangle into five equal parts.

5. Inscribe a circle in a sector of a given circle.

6. The line DE is parallel to the base BC of the triangle ABC: prove that the circles described about the triangles ABC, ADE touch at A.

7. The diagonals of a cyclic quadrilateral intersect in E: prove that the tangent at E to the circle about the triangle ABE is parallel to CD.

8. Inscribe a regular octagon in a given square.

9. A line of given length slides between two given lines: find the locus of the intersection of perpendiculars from its extremities to the given lines.