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has three equal sides.
equal, and all its angles right angles.
has not all its sides equal.
are not right angles.
another, but all its sides are not equal, nor its angles right
which, being produced ever so far both ways, do not meet.
any one point to any other point.
in a straight line.
distance from that centre.
Things which are halves of the same thing are equal to one
VIII. Magnitudes which coincide with one another, that is, which exactly fill the same space, are equal to one another.
IX. The whole is greater than its part.
X. All right angles are equal to one another. “Two straight lines, which intersect one another, cannot be
“ both parallel to the same straight line.”
PROPOSITION I. PROBLEM.
TO describe an equilateral triangle upon a given finite straight line.
Let AB be the given straight line; it is required to describe an equilateral triangle upon it.
From the centre A, at
the distance AB, desa 3. Postu- scribea the circle BCD, late.
and from the centre B, at
another, draw the straight b 1. Post. linesb CA, CB, to the
points A, B: ABC shall
Because the point X is the centre of the circle BCD, AC c 11. Defi- is equal to AB; and because the point B is the centre of the
circle ACE, BC is equal to BA: but it has been proved that
equal to AB; but things which are equal to the same are d 1. Axi- equal to one anotherd; therefore CA is equal to CB; where
fore CA, AB, BC are equal to one another; and the triangle
PROP. II. PROB.
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.