SEPTEMBER 1890. PLANE. 1. Define locus. Find the locus of all points equidistant from two parallel lines, proving result. 2. The common chord of two circles is perpendicular to, and is bisected by, the line joining their centers. 3. To divide a given line in extreme and mean ratio.What regular inscribed polygons may be constructed by means of this division? Prove your statement. 4. Two circles are tangent internally, the ratio of their radii being 3 Compare their areas, and also the area left in the larger circle with each. 5. Of all polygons formed of given sides the maximum may be inscribed in a circle. SEPTEMBER 1890. SOLID. 1. The intersections of two parallel planes by a third plane are parallel lines. 2. Two tetrahedrons having a trihedral angle equal, are as the products of the including edges. 3. In any spherical triangle, the greater angle is opposite the greater side, and conversely. 4. A certain spherical triangle is in area, one-fifth that of its sphere. If the relation connecting the angles be 2A 3B6C, find A, B, and C. JUNE 1891. PLANE. 1. In what various ways may systems of three lines be drawn similarly in a triangle to meet in one point? Prove any one of your statements. 2. How is an angle between two secants to a circle measured? between a tangent and a chord? Give demonstrations of answers. 3. If through any point within a circle a chord be drawn, the product of the segments formed by the point is constant in whatever direction the chord be drawn. 4. To construct a square whose area is double that of a given square. 5. Regular polygons of the same number of sides are similar. SEPTEMBER 1891. PLANE. 1. Mutually equilateral triangles are equal in all respects. 2. The common chord of two circles is bisected perpendicularly by the line of centers. What does this proposition become when the circles touch? 3. State and prove the theorems possible when a perpendicular is let fall upon the hypothenuse of a right angled triangle from the opposite vertex. 4. Compute the area of a regular hexagon whose side is 5 feet. Construct a triangle of equivalent area. 5, The square on the side opposite any acute angle of a triangle is equivalent to the sum of the squares on the other two sides diminished by twice the rectangle on one of those sides and the projection of the other upon it. JUNE 1891. SOLID. 1. All perpendiculars to a line at any one point lie in a plane perpendicular to that line at that point. 2. A line and plane perpendicular to the same line are parallel, or the plane contains the line. 3. The sum of two face angles of a trihedral angle is greater than the third. 4. State and prove the two propositions regarding first the volume of a triangular pyramid, and second, the volume of any pyramid. 5. Give a method of calculating the area of any spherical triangle, and apply it to an equiangular triangle whose sides are arcs of 60° on a sphere of unit radius. |