The Elements of Euclid, books i-vi; xi. 1-21; xii. 1,2; ed. by H.J. Hose, Book 1 |
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Page 16
... proved above that angle FBC is equal to angle GCB . Therefore ( 1 ) the angles ABC , ACB at the base BC are equal ; ( 2 ) the angles CBD , BCE on the other side of the base BC are equal . Which was to be proved . COR . - Every ...
... proved above that angle FBC is equal to angle GCB . Therefore ( 1 ) the angles ABC , ACB at the base BC are equal ; ( 2 ) the angles CBD , BCE on the other side of the base BC are equal . Which was to be proved . COR . - Every ...
Page 17
... proved . COR . - Every equiangular triangle shall also be equi- lateral . Let ABC be an equiangular tri- angle . Then it shall also be equi- lateral . Since the triangle ABC is equian- gular , the angle ABC is equal to the angle ACB ...
... proved . COR . - Every equiangular triangle shall also be equi- lateral . Let ABC be an equiangular tri- angle . Then it shall also be equi- lateral . Since the triangle ABC is equian- gular , the angle ABC is equal to the angle ACB ...
Page 19
... proved . PROP . VIII . THEOR . If two triangles have the three sides of the one respectively equal to the three sides of the other : then these triangles shall be equal in every respect , i . e . ( 1 ) the three angles of the one shall ...
... proved . PROP . VIII . THEOR . If two triangles have the three sides of the one respectively equal to the three sides of the other : then these triangles shall be equal in every respect , i . e . ( 1 ) the three angles of the one shall ...
Page 26
... proved that no other straight line through B on the other side of AB can be in the same straight line with CB but BD . Therefore BD is in the same straight line with CB ; or CB , DB are in one straight line . Which was to be proved ...
... proved that no other straight line through B on the other side of AB can be in the same straight line with CB but BD . Therefore BD is in the same straight line with CB ; or CB , DB are in one straight line . Which was to be proved ...
Page 27
... proved . COR . 1. — If two straight lines cut one another : then the four angles which they make at the point of intersection shall be together equal to four right angles . Let AB , CD cut one another in E. Then the four angles AEC ...
... proved . COR . 1. — If two straight lines cut one another : then the four angles which they make at the point of intersection shall be together equal to four right angles . Let AB , CD cut one another in E. Then the four angles AEC ...
Common terms and phrases
ABC is equal adjacent angles angle ABC angle ACB angle BAC angle DEF angle EDF angles are equal angular points base BC bisected centre circle ABC circle DEF circumference const cut the circle diagonal draw equal angles equal Ax equals add equi equimultiples exterior angle FGHKL fore four magnitudes fourth given straight line gnomon hence the angle homologous hyps included angle interior and opposite join less Let ABC multiple opposite angle parallelogram perpendicular produced PROP proportional proved radius rectangle contained remaining angle respectively equal right angles segment shewn sides BC similar solid angle square of AC THEOR thing are equal third angle three angles touching the circle triangle ABC unequal
Popular passages
Page 39 - To draw a straight line through a given point parallel to a given straight line. Let A be the given point, and BC the given straight line...
Page 52 - To a given straight line to apply a parallelogram, which shall be equal to a given triangle, and have one of its angles equal to a given rectilineal angle.
Page 16 - To draw a straight line at right angles to a given straight line, from a given point in the same.
Page 5 - From a given point to draw a straight line equal to a given straight line. Let A be the given point, and BC the given straight line ; it is required to draw, from the point A, a straight line equal to BC.
Page 11 - Upon the same base, and on the same side of it, there cannot be two triangles that have their sides which are terminated in one extremity of' the base, equal to one another, and likewise those which are terminated in the other extremity.
Page 105 - If a straight line touch a circle, and from the point of contact a straight line be drawn cutting the circle ; the angles which this line makes with the line touching the circle, shall be equal to the angles which are in the alternate segments of the circle.
Page 118 - 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 66 - If a straight line be divided into two equal parts, and also into two unequal parts; the rectangle contained by the unequal parts, together with the square of the line between the points of section, is equal to the square of half the line.
Page 6 - From the greater of two given straight lines to cut off a part equal to the less. Let AB and C be the two given straight lines, whereof AB is the greater.
Page 119 - Upon a given straight line to describe a segment of a circle, which shall contain an angle equal to a given rectilineal angle.