Elements of geometry, containing the first two (third and fourth) books of Euclid, with exercises and notes, by J.H. Smith, Part 21872 |
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Page 142
... TANGENT to , or to touch , a circle , when it meets and , being produced , does not cut the circle . From this definition it follows that the tangent meets the circle in one point only , for if it met the circle in two points it would ...
... TANGENT to , or to touch , a circle , when it meets and , being produced , does not cut the circle . From this definition it follows that the tangent meets the circle in one point only , for if it met the circle in two points it would ...
Page 143
... tangent to the circle . D B E Let ABC be a O , of which the centre is 0 , and the diameter AOB . Draw DE at right angles to AOB . Then must DE be a tangent to the . Take any point P in DE , and join OP . Then , OBP is a right angle ...
... tangent to the circle . D B E Let ABC be a O , of which the centre is 0 , and the diameter AOB . Draw DE at right angles to AOB . Then must DE be a tangent to the . Take any point P in DE , and join OP . Then , OBP is a right angle ...
Page 144
... tangent to O BCD . Next , let the given pt . be on the ○ ce of the © , as B. Then , if BA be drawn to the radius OB , BA is a tangent to the C at B. I. 16 . Q. E. D. Ex . 1. Shew that the two tangents drawn from a point without the ...
... tangent to O BCD . Next , let the given pt . be on the ○ ce of the © , as B. Then , if BA be drawn to the radius OB , BA is a tangent to the C at B. I. 16 . Q. E. D. Ex . 1. Shew that the two tangents drawn from a point without the ...
Page 162
... tangents is double of the angle contained by the line joining the points of contact , and the diameter drawn through one of them . Ex . 4. The vertical angle of any oblique - angled triangle inscribed in a circle is greater or less than ...
... tangents is double of the angle contained by the line joining the points of contact , and the diameter drawn through one of them . Ex . 4. The vertical angle of any oblique - angled triangle inscribed in a circle is greater or less than ...
Page 163
... : . LDFA = L FED , that is , ¿ DFA = △ in segment FED . Q. E.D. Ex . The chord joining the points of contact of parallel tangents is a diameter . PROPOSITION XXXIII . PROBLEM . On a given straight line BOOK III . PROP . XXXII . 163.
... : . LDFA = L FED , that is , ¿ DFA = △ in segment FED . Q. E.D. Ex . The chord joining the points of contact of parallel tangents is a diameter . PROPOSITION XXXIII . PROBLEM . On a given straight line BOOK III . PROP . XXXII . 163.
Common terms and phrases
ABCD angular points bisect the angle centre of ADE chord AC circle be described circle described circles cut circles intersect circles touch circumference coincide cutting the circle Describe a circle diagonals diameter draw equal circles equiangular equilateral triangle given circle given line given point given square given straight line isosceles triangle Join OA Let ABC line be drawn line drawn meet the Oce middle points opposite sides parallel parallelogram pass perpendicular point of contact produced prove Q. E. D. Ex Q. E. F. PROPOSITION quadrilateral quadrilateral figure rect rectangle contained reflex angle regular pentagon regular polygon required to inscribe rhombus right angles segment ABC semicircle shew shewn straight line joining subtended sum of 48 tangents THEOREM touch the circle triangle ABC vertex
Popular passages
Page 152 - The opposite angles of any quadrilateral figure inscribed in a circle, are together equal to two right angles.
Page 168 - 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.
Page 161 - The angle in a semicircle is a right angle; the angle in a segment greater than a semicircle is less than a right angle; and the angle in a segment less than a semicircle is greater than a right angle.
Page 132 - If a point be taken within a circle, from which there fall more than two equal straight lines to the circumference, that point is the centre of the circle. Let...
Page 163 - IF a straight line touch a circle, and from the point of contact a straight line be drawn at right angles to the touching line, the centre of the circle shall be in that line.
Page 177 - IN a given circle to inscribe a triangle equiangular to a given triangle. Let ABC be the given circle, and DEF the given triangle; it is required to inscribe in the circle ABC a triangle equiangular to the triangle DEF. Draw (1 7. 3.) the straight line G AH touching the circle in the point A, and. at the point A, in the straight line AH, make (23.
Page 184 - ABD is described, having each of the angles at the base double of the third angle.
Page 207 - Upon the same straight line, and upon the same side of it, there cannot be two similar segments of circles, not coinciding with one another. If it be possible. let the two similar segments of circles, viz. ACB' ADB be upon the same side of the same straight line AB, not coinciding with one another.
Page 203 - If a straight line be divided into any two parts, the squares on the whole line, and on one of the parts, are equal to twice the rectangle contained by the whole and that part, together with the square on the other part.
Page 183 - To inscribe a circle in a given square. Let ABCD be the given square ; it is required to inscribe a circle in ABCD.