# Calendar: 1896-1897 Page 692

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Cli FACULTY OF SCIENCE prove that one of the three joining lines is equal to the other two together 11 If straight line is perpendicular to each of two inter- sccting straight lines at their point of intersection it is perpendicular to the plane which contains them If point outside plane is equidistant from three points in the plane it is also at the same distance from any point of the circumference of the circle passing through those three points 12 Shew how to draw straight line perpendicular to given plane from given point without it Prove that the perpendicular from the vertex of regular tetrahedron upon the opposite face is three times that dropped from its intersection with that face upon any one of the other faces tomctvi antl Differential CalcuTusf YEAR III 13 If triangle be inscribed in circle the sum of the squares on its sides is less than equal to or greater than twice the square on the diameter of the circle according as the triangle is obtuse-angled right-angled or acute- angled The sum of the squares on the sides is greatest if the triangle be equilateral In the last case compare the sum of the squares on the sides and the square on the diameter 14 The foot of the perpendicular from the focus on the tangent at any point Ρ of parabola lies on the tangent at the vertex and the perpendicular is mean proportional between SP and SA If Fbe the perpendicular from the vertex upon the tangent at Ρ to the parabola focus and if PN be the ordinate to the axis prove that AY2 AN2 AS SP 15 In an ellipse the rectangles contained by the segments of any two chords which intersect each other are in the ratio of the squares of the parallel semi-diameters If circle cut an ellipse at two points and touch it at third point shew that the common tangent and common chord are equally inclined to the axis

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