Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/416

 552. Find the point in the circumference of a given circle, the sum of whose distances from two given straight lines at right angles to each other, which do not cut the circle, is the greatest or least possible.

553. On the sides of a triangle segments of a circle are described internally, each containing an angle equal to the excess of two right angles above the opposite angle of the triangle: shew that the radii of tho circles are equal, that the circles all pass through one point, and that their chords of intersection are respectively perpendicular to the opposite sides of the triangle.

IV. 1 to 16.

554. From the angles of a triangle ABC perpendiculars are drawn to the opposite sides meeting them at D, E, F respectively: shew that DE and DF are equally inclined to AD.

555. The points of contact of the inscribed circle of a triangle are joined; and from tho angular points of the triangle so formed perpendiculars arc drawn to tho opposite sides: shew that the triangle of which the feet of these perpendiculars are the angular points has its sides parallel to the sides of the original triangle.

556. Construct a triangle having given an angle and the radii of the inscribed and circumscribed circles.

557. Triangles are constructed on the same baSe with equal vertical angles; shew that the locus of the centres of the escribed circles, each of which touches one of the sides externally and the other side and base produced, is an arc of a circle, the centre of which is on the circumference of the circle circumscribing the triangles.

558. From the angular points A, B, C of a triangle perpendiculars are drawn on the opposite sides, and terminated at the points D, E, F on the circumfercnce of the circumscribing circle: if L be the point of intersection of the perpendiculars, shew that LD, LE, LF are bisected by the sides of the triangle.