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 166
... rect . AP , PC = rect . BP , PD . From O , the centre , draw OM , ON 18 to AC , BD , and join OA , OB , OP . Then AC is divided equally in M and unequally in P , . : rect . AP , PC with sq . on MP = sq . on AM . Adding to each the sq ...
... rect . AP , PC = rect . BP , PD . From O , the centre , draw OM , ON 18 to AC , BD , and join OA , OB , OP . Then AC is divided equally in M and unequally in P , . : rect . AP , PC with sq . on MP = sq . on AM . Adding to each the sq ...
Page 167
... rect . AD , DB with sq . on OB = sq . on OD " ; .. rect . AD , DB with sq . on OB = sqq . on OC , DC . L. 47 . And sq . on OB = sq . on OC ; .. rect . AD , DB = sq . on DC . Q. E. D. Ex . 1. Two circles intersect in A and B ; shew that ...
... rect . AD , DB with sq . on OB = sq . on OD " ; .. rect . AD , DB with sq . on OB = sqq . on OC , DC . L. 47 . And sq . on OB = sq . on OC ; .. rect . AD , DB = sq . on DC . Q. E. D. Ex . 1. Two circles intersect in A and B ; shew that ...
Page 168
... rect . DA , AC = sq . on AE . But rect . DA , AC = sq . on AB ; .. sq . on AB = sq . on AE ; .. AB = AE . Then in the As OAB , OAE , * OB = OE , and OA is common , and AB = AE , .LABO = = LAEO . III . 36 . Нур . L. C. But AEO is a rt ...
... rect . DA , AC = sq . on AE . But rect . DA , AC = sq . on AB ; .. sq . on AB = sq . on AE ; .. AB = AE . Then in the As OAB , OAE , * OB = OE , and OA is common , and AB = AE , .LABO = = LAEO . III . 36 . Нур . L. C. But AEO is a rt ...
Page 184
... rect . AB , BC = sq . on AC , and BD = AC , . : rect . AB , BC = sq . on BD , and .. BD touches the ACD . III . 37 . Then BD touches ACD , and DC is a chord of the C , .. LBDC = L CAD . Add to each △ CDA . Then △ BDA = sum of 4 8 CAD ...
... rect . AB , BC = sq . on AC , and BD = AC , . : rect . AB , BC = sq . on BD , and .. BD touches the ACD . III . 37 . Then BD touches ACD , and DC is a chord of the C , .. LBDC = L CAD . Add to each △ CDA . Then △ BDA = sum of 4 8 CAD ...
Page 204
... rect- angles contained by BC and DK , CA and EL , AB and FM is equal to the sum of the other two . m . 35. Through a point , within a circle , draw a chord , such that the rectangle contained by the whole chord and one part may be equal ...
... rect- angles contained by BC and DK , CA and EL , AB and FM is equal to the sum of the other two . m . 35. Through a point , within a circle , draw a chord , such that the rectangle contained by the whole chord and one part may be equal ...
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.