Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/389

 by the part of it without the circle and the part within the circle may be equal to a given square not greater than that on the diameter.

IV. 1 to 4.

278. In IV. 3 shew that the straight lines drawn through A and B to touch the circle will meet.

279. In IV. 4 shew that the straight lines which bisect the angles B and C will meet.

280. In IV. 4 shew that the straight line DA will bisect the angle at A.

281. If the circle inscribed in a triangle ABC touch the sides AB,AC at the points D, E, and a straight line be drawn from A to the centre of the circle meeting the circumference at G, show that the point G is the centre of the circle inscribed in the triangle ADE.

282. Shew that the straight lines joining the centres of the circles touching one side of a triangle and the others produced, pass through the angular points of the triangle.

283. A circle touches the side BC of a triangle ABC and the other two sides produced: shew that the distance between the points of contact of the side BG with this circle and with the inscribed circle, is equal to the difference between the sides AB and AC.

284. A circle is inscribed in a triangle ABC and a triangle is cut off at each angle by a tangent to the circle. Shew that the sides of the three triangles so cut off are together equal to the sides of ABC.

285. D is the centre of the circle inscribed in a triangle BAC and AD produced to meet the straight line drawn through B at right angles to BD at O: shew that O is the centre of the circle which touches the side BC and the sides AB, AC produced.

286. Three circles are described, each of which touches one side of a triangle ABC, and the other two sides produced. If D be the point of contact of the side BC, E that of AC, and F that of AB, shew that AE is equal to BD, BF to CE, and CD to AF.

287. Describe a circle which shall touch a given circle and two given straight lines which themselves touch the given circle.