Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/391

 300. Describe a circle which shall touch a given straight line, and pass through two given points.

301. Describe a circle which shall pass through two given points and cut off from a given straight line a chord of given length.

302. Describe a circle which shall have its centre in a given straight lino, and cut off from two given straight lines chords of equal given length.

303. Two triangles have equal bases and equal vertical angles: shew that the radius of the circumscribing circle of one triangle is equal to that of the other.

304. Describe a circle which shall pass through two given points, so that the tangent drawn to it from another given point may be of a given length.

305. C is the centre of a circle; CA, CB are two radii at right angles; from B any chord BP is drawn cutting CA at N: a circle being described round ANP, shew that it will be touched by BA.

306. AB and CD are parallel straight lines, and the straight lines which join their extremities intersect at E: shew that the circles described round the triangles ABE, CDE touch one another.

307. Find the centre of a circle cutting off three equal chords from the sides of a triangle.

308. If O be the centre of the circle inscribed in the triangle ABC, and AO be produced to meet the circumscribed .circle at F shew that FB, FO, and FC are all equal.

309. The opposite sides of a quadrilateral inscribed in a circle are produced to meet at P and Q, and about the triangles so formed without the quadrilateral, circles are described meeting again at R: shew that P, R, Q are in one straight line.

310. The angle ACB of any triangle is bisected, and the base AB is bisected at right angles, by straight lines which intersect at D: shew that the angles ACB, ADB are together equal to two right angles.

311. ACDB is a semicircle, AB being the diameter, and the two chords AD, BC intersect at E: shew that it a circle be described round CDE it will cut the former at right angles.