Elements of plane geometry, book i, containing nearly the same propositions as the first book of Euclid's Elements |
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Seite 27
He proposes to prove , that “ If the angles at the base of a triangle are equal , the sides opposite to them are also equal . " names the sides opposite the equal angles AB , AC , magnitudes which may evidently assume three states ...
He proposes to prove , that “ If the angles at the base of a triangle are equal , the sides opposite to them are also equal . " names the sides opposite the equal angles AB , AC , magnitudes which may evidently assume three states ...
Seite 40
... of BA and AC , is greater than the straight line BC , and , for the same reason , AB and BC are greater than AC , and AC and CB greater than AB ; and therefore any two sides of a triangle are together greater than the third side .
... of BA and AC , is greater than the straight line BC , and , for the same reason , AB and BC are greater than AC , and AC and CB greater than AB ; and therefore any two sides of a triangle are together greater than the third side .
Seite 42
On the same base , and on the same side of it , there cannot be two triangles which have their two sides which are ... Let ABC and DEF be two triangles which have the side AB equal to DE , AC to DF , and BC to EF , the triangles are ...
On the same base , and on the same side of it , there cannot be two triangles which have their two sides which are ... Let ABC and DEF be two triangles which have the side AB equal to DE , AC to DF , and BC to EF , the triangles are ...
Seite 43
A A G C E F so that the point B be on E , and the straight line BC on EF , the G point C shall coincide with the point F , because BC is equal to EF ; the side BC therefore coinciding with EF , the sides BA B and AC shall coincide with ...
A A G C E F so that the point B be on E , and the straight line BC on EF , the G point C shall coincide with the point F , because BC is equal to EF ; the side BC therefore coinciding with EF , the sides BA B and AC shall coincide with ...
Seite 44
D NÁ B F to DE ; and AC shall coincide A with DF , because the angle A is equal to D ; and the point C shall ... Because in the triangles ADF and AEF , the side AD is equal to AE , the B side FD to FE , and AF is common to both F ...
D NÁ B F to DE ; and AC shall coincide A with DF , because the angle A is equal to D ; and the point C shall ... Because in the triangles ADF and AEF , the side AD is equal to AE , the B side FD to FE , and AF is common to both F ...
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Elements of Plane Geometry, Book: Containing Nearly the Same Propositions As ... Euclid Keine Leseprobe verfügbar - 2008 |
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ABC is equal AC is equal acute adjacent alternate angles ancient angle ACD angle BAC angles ABC appears application assume axiom base BC bisect called centre circle circumference coincide common conclusion construction definition demonstration describe determined diagonal draw drawn elements employed established Euclid extended exterior angle extremities fall four right angles geometers geometry given straight line greater half Hence included angle interior opposite angle intersect introduced join knowledge less Let ABC magnitudes manner means meet method mind mode necessary obtuse parallel lines parallelogram perpendicular plane position principle problem produced proof properties PROPOSITION proved radiant reason rectangle rectilineal figure remaining respects side AB side AC surfaces THEOR thing third triangle ABC triangles are equal truths unequal vertex wherefore whole
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Seite 43 - If two triangles have two sides and the included angle of the one, equal to two sides and the included angle of the other, each to each, the two triangles will be equal.
Seite 46 - Any two angles of a triangle are together less than two right angles.
Seite 37 - The angles which one straight line makes with another upon one tide of it, are either two right angles, or are together equal to two right angles. Let the straight line AB make with CD, upon one side of it the angles CBA, ABD ; these are either two right angles, or are together equal to two right angles. For, if the angle CBA be equal to ABD, each of them is a right angle (Def.
Seite 57 - Through a given point to draw a straight line parallel to a given straight line, Let A be the given point, and BC the given straight line : it is required to draw through the point A a straight line parallel to BC.
Seite 38 - ... in one and the same straight line. At the point B in the straight line AB, let the two straight lines BC, BD upon the opposite sides of AB, make the adjacent angles ABC, ABD, equal together to two right angles. BD is in the same straight line with CB.
Seite 68 - All the interior angles of any rectilineal figure, together with four right angles, are equal to twice as many right angles as the figure has sides.
Seite 34 - LET it be granted that a straight line may be drawn from any one point to any other point.
Seite 64 - Parallelograms upon the same base, and between the same parallels, are equal to one another.
Seite 46 - If one side of a triangle be produced, the exterior angle is greater than either of the interior, and opposite angles.
Seite 34 - Things which are equal to the same thing are also equal to one another. 2. If equals be added to equals, the wholes are equal. 3. If equals be subtracted from equals, the remainders are equal.