Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/394

 333. In the figure of IV. 10 if AF the diameter of the smaller circle, DF is equal to a radius of the circle which circumscribes the triangle BCD.

IV. 11 to 16.

334. The straight lines which connect the angular points of a regular pentagon which are not adjacent intersect at the angular points of another regular pentagon.

335. ABCDE is a regular pentagon; join AC and BE, and let BE meet AC at F; shew that AC is equal to the sum of AB and BF.

336. Shew that each of the triangles made by joining the extremities of adjoining sides of a regular pentagon is less than a third and greater than a fourth of the whole area of the pentagon.

337. Shew how to derive a regular hexagon from an equilateral triangle inscribed in a circle, and from the construction shew that the side of the hexagon equals the radius of the circle, and that the hexagon is double of the triangle.

338. In a given circle inscribe a triangle whose angles are as the numbers 2, 5, 8.

339. If ABCDEF is a regular hexagon, and AC, BD, CE, DF, EA, FB be joined, another hexagon is formed whose area is one third of that of the former.

340. Any equilateral figure which is inscribed in a circle is also equiangular.

VI. 1,2.

341. Shew that one of the triangles in the figure of IV. 10 is a mean proportional between the other two.

342. Through D, any point in the base of a triangle ABC, straight lines DE, DF are drawn parallel to the sides AB, AC, and meeting the sides at E, F: shew that the triangle AEF a mean proportional between the triangles FBD, EDC.