Calendar: 1898-1899 Page 744

CX11 FACULTY OF SCIENCE A2B2C2 is obtained in like manner and so on Find the angles of the triangle AnBnCn and prove that when η in- creases without limit the triangles become equiangular euclfo YL XI YEARS II III 10 ABCD is cyclic quadrilateral AC being diameter of the circumscribing circle AD BC meet in and AB DC in prove that EF EC ED FC FB 11 If the vertical angle of triangle be bisected by straight line which also cuts the base the segments of the base shall bear to one another the ratio of the remaining sides of the triangle Also state and prove the converse theorem If The the centre of the inscribed circle of triangle and if AI meet BC in prove that IX AX BC BC CA AB 12 Triangles which are equal in area and which have one angle of the one equal to one angle of the other have their sides about the equal angles reciprocally proportional From triangle ABC cut off triangle EBF such that the side EF is parallel to given line and the triangle EBF is one-third of the triangle ABC 13 Define and prove the existence of the nine-point circle of triangle Shew that the part of the base intercepted by it subtends at the circumference of the circle an angle equal to the difference of the base angles of the triangle 14 In any triangle ABC if AD BE CF be the perpen- dicular from the angular points on the opposite sides and OX ΟΥ OZ the perpendiculars on the corresponding sides from any point prove that OX OY OZ AD BE CF 15 Shew that if two lines intersect one another they are in plane Shew that no two straight lines joining different pairs of points in two non-intersecting straight lines can cut one another