*EXERCISES IN EUCLID*.

288. If the three points be joined in which the circle inscribed in a triangle meets the sides, shew that the resuiting 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 bo 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*.