## Mathematical Questions and Solutions, from the "Educational Times.", Volume 3 |

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angle asymptotes axes axis becomes centre chord circle circumscribing coincide common conic conjugate constant construction coordinates corresponding curve denote describe determined diameter difference double draw drawn ellipse envelope equal equation evidently expression figure fixed points follows four given line given points harmonic hence hyperbola infinity inscribed intersection involution joining known latter locus meet middle points obtain opposite pairs parallel pass perpendiculars plane points of contact polar position problem produced Proposed prove quadrilateral Question radius reference relative respectively result segments semicircle sides similar similarly sinē Solution square straight line suppose taking tangent theorem third touch triangle ABC vertices whence

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Page 16 - The line joining the middle points of two sides of a triangle is parallel to the third side and equal to half of the third side.

Page 41 - IF a straight line be divided into two equal, and also into two unequal parts ; the squares of the two unequal parts are together double of the square of half the line, and of the square of the line between the points of section.

Page v - There are n points in a plane, no three of which are in the same straight line with the exception of p, which are all in the same straight line; find the number of lines which result from joining them.

Page 71 - CE is equal to the difference of the segments of the base made by the perpendicular.

Page xii - Construct a triangle, having given the base, tho vertical angle, and the length of the straight line drawn from the vertex to the base bisecting the vertical angle. 551. A, B, C are three given points in the circumference of a given circle : find a point P such that if AP, BP, CP meet the circumference at D, E, F respectively, the arcs DE, EF may be equal to given arcs. 552. Find...

Page xv - If from the intersection of the diagonals of a quadrilateral inscribed in a circle perpendiculars be...

Page 63 - Cc in r; and so on. Prove that, after going twice round the triangle in this way, we always come back to the same point. Show that the theorem is its own reciprocal. Find the analogous properties of a skew quadrilateral in space, and of a polygon of n sides in a plane. Solution by PBOFESSOE CAYLEY.

Page 23 - Let nCr denote the number of combinations of n things taken r at a time.

Page 90 - A-sin2 o) = &c. ; and its quasi-reciprocal and polar, cos b cos c, cos b, cos c cos a, cos c cos a, cos c cos a, cos b, cos a cos...

Page 8 - PROBLEM VIII. It is required to Investigate a Theorem, by means of which, Spherical Triangles, whose Sides are Small compared with the radius, may be solved by the rules for Plane Trigonometry, without considering the Chords of the respective Arcs or Sides.