Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/390

 288. If the three points be joined in which the circle inscribed in a triangle meets the sides, shew that the resulting triangle is acute angled.

289. Two opposite sides of a quadrilateral are together equal to the other two, and each of the angles is less than two right angles. Shew that a circle can be inscribed in the quadrilateral.

290. Two circles HPL, KPM, that touch each other externally, have the common tangents HK, LM; HL and KM being joined, shew that a circle may be inscribed in the quadrilateral HKML.

291. Straight lines are drawn from the angles of a triangle to the centres of the opposite described circles: shew that these straight lines intersect at the centre of the inscribed circle.

292. Two sides of a triangle whose perimeter is constant are given in position; shew that the third side always touches a certain circle.

293. Given the base, the vertical angle, and the radius of the inscribed circle of a triangle, construct it.

IV. 5 to 9.

294. In IV. 5 shew that the perpendicular from F on BC will bisect BC.

295. If DE be drawn parallel to the base BC of a triangle ABC, shew that the circles described about the triangles ABC and ADE have a common tangent.

296. If the inscribed and circumscribed circles of a triangle be concentric, shew that the triangle must be equilateral.

297. Shew that if the straight line joining the centres of the inscribed and circumscribed circles of a triangle passes through one of its angular points, the triangle is isosceles.

298. The common chord of two circles is produced to any point P; PA touches one of the circles at A, PBC is any chord of the other. Shew that the circle which passes through A, B, and C touches the circle to which PA is a tangent.

299. A quadrilateral ABCD is inscribed in a circle, and AD, BC are produced to meet at E: shew that the circle described about the triangle ECD will have the tangent at E parallel to AB.