The Solutions of the Geometrical Problems: Consisting Chiefly of Examples in Plane Co-ordinate Geometry, Proposed at St. John's College, Cambridge, from Dec. 1830 to Dec. 1846. With an Appendix, Containing Several General Properties of Curves of the Second Order, and the Determination of the Magnitude and Position of the Axes of the Conic Section Represented by the General Equation of the Second DegreeJ. & J. J. Deighton, 1847 - 263 páginas |
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Página 1
... bisected by the cir- cumference . 4. The line joining the centres of the inscribed and cir- cumscribed circles of a triangle subtends at any one of the angular points an angle equal to the semidifference of the other two angles . 5 ...
... bisected by the cir- cumference . 4. The line joining the centres of the inscribed and cir- cumscribed circles of a triangle subtends at any one of the angular points an angle equal to the semidifference of the other two angles . 5 ...
Página 4
... bisect it in ; then CP is the diameter , and O the centre of the inner circle PQC when P becomes the point of contact . Let the circle PQC cut AC in Q , and join OQ ; then PQ 2 PA △ POQ = 2 4 PCQ , or ; PO PC ... arc PQ 2PO PC arc PA ...
... bisect it in ; then CP is the diameter , and O the centre of the inner circle PQC when P becomes the point of contact . Let the circle PQC cut AC in Q , and join OQ ; then PQ 2 PA △ POQ = 2 4 PCQ , or ; PO PC ... arc PQ 2PO PC arc PA ...
Página 5
... bisect DP in E , and draw DF , EF respectively perpendicular to AB , DP meeting each other in F ; with centre F and distance FD describe a circle ; this will pass through the point P and touch the straight line AB ; and △ PDB = the ...
... bisect DP in E , and draw DF , EF respectively perpendicular to AB , DP meeting each other in F ; with centre F and distance FD describe a circle ; this will pass through the point P and touch the straight line AB ; and △ PDB = the ...
Página 8
... bisected in C ; and in like manner it may be proved that a , b are the middle points of BC , CA respectively . Draw PM parallel to AC , then if AB , AC be taken for the co - ordinate axes , and x , y be the co- ordinates of P ; PM : AC ...
... bisected in C ; and in like manner it may be proved that a , b are the middle points of BC , CA respectively . Draw PM parallel to AC , then if AB , AC be taken for the co - ordinate axes , and x , y be the co- ordinates of P ; PM : AC ...
Página 10
... bisected by the line joining it with the other extremity . 5. The lines which bisect the vertical angles of all triangles on the same base and with the same vertical angle , all intersect in one point . 6. If a semicircle be described ...
... bisected by the line joining it with the other extremity . 5. The lines which bisect the vertical angles of all triangles on the same base and with the same vertical angle , all intersect in one point . 6. If a semicircle be described ...
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Términos y frases comunes
2c sin² a₁ a²² ABCD angular points asymptotes axes axis axis-major ay² b₁ bisected C₁ chord of contact co-ordinates conic section conjugate conjugate hyperbola constant cos² cosw curve diagonals diameter draw ellipse equal Euclid find the locus fixed point given points given straight lines hence hk g hyperbola inclined inscribed joining the points latus rectum Let A fig line joining m₁ meet middle point n₂ pair of tangents parabola parallel parallelogram pass through three perpendicular plane points of contact points of intersection polar equation polygon position quadrilateral figure radius remaining sides respectively right angles shew Similarly sin w sin² ST JOHN'S COLLEGE t₁ t₂ tangents be drawn tangents drawn three sides touch vertex y₁
Pasajes populares
Página 54 - If two triangles which have two sides of the one proportional to two sides of the other, be joined at one angle, so as to have their homologous sides parallel to one another ; the remaining sides shall be in a straight line. Let ABC, DCE be two triangles which have the two sides BA, AC proportional to the two CD, DE, viz.
Página 117 - Similar triangles are to one another in the duplicate ratio of their homologous sides.
Página 117 - IF from any point without a circle two straight lines be drawn, one of which cuts the circle, and the other touches it ; the rectangle contained by the whole line which cuts the circle, and the part of it without the circle,. shall be equal to the square of the line which touches it.
Página 96 - The rectangle contained by the diagonals of a quadrilateral ,figure inscribed in a circle, is equal to both the rectangles contained by i'ts opposite sides.
Página 16 - MAGNITUDES which have the same ratio to the same magnitude are equal to one another ; and those to which the same magnitude has the same ratio are equal to one another.
Página 28 - To divide a given straight line into two parts, so that the rectangle contained by the whole and one part may be equal to the square on the other part*.
Página 28 - THEOREM. lf the first has to the second the same ratio which the third has to the fourth, but the third to the fourth, a greater ratio than the fifth has to the sixth ; the first shall also have to the second a greater ratio than the fifth, has to the sixth.
Página 10 - ... not in the same plane with the first two ; the first two and the other two shall contain equal angles.
Página 87 - The locus of the middle points of a system of parallel chords in a parabola is called a diameter.