Euclid Book 1
Book 1 definitions include that of lines, points, circles, planes, triangles, angles and other geometric figures. The propositions in Book 1 use the definitions, postulates and common notions to develop important topics related to plane geometry such as triangle congruence theorems (Propositions 4, 8, 26 a and 26 b), the triangle inequality (Proposition 20) and the Pythagorean theorem (Proposition 47).
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Euclid Book 1 Proposition 1
To construct an equilateral triangle on a given line segment.
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Euclid Book 1 Proposition 2
From a given point to draw a line segment equal to a given line segment.
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Euclid Book 1 Proposition 3
To cut off from the longer of two given unequal line segments a part equal to the shorter.
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Euclid Book 1 Proposition 4
If two triangles have two sides of one equal respectively to two sides of the other, and have also the angles included by those sides equal, the triangles shall be congurent.
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Euclid Book 1 Proposition 5
The angles at the base of an isosceles triangle are equal to one another, and if the equal sides are produced, the external angles below the base will be equal.
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Euclid Book 1 Proposition 6
If in a triangle, two angles are equal, then the sides opposite to them are also equal.
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Euclid Book 1 Proposition 7
If two triangles on the same base and on the same side of it have one pair of conterminous sides equal to one another, then the other pair of conterminous sides must be unequal.
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Euclid Book 1 Proposition 8
If two triangles have two sides of one respectively equal to two sides of the other, and have also the base of one equal to the base of the other, then the two triangles are congurent.
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Euclid Book 1 Proposition 9
To bisect a given angle.
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Euclid Book 1 Proposition 10
To bisect a given line segment.
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Euclid Book 1 Proposition 11
From a given point in a given line segment to draw a straight line perpendicular to the given line.
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Euclid Book 1 Proposition 12
To draw a line perpendicular to a given infinite line from a given point not on it.
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Euclid Book 1 Proposition 13
One line standing on another line will make either two right angles or two adjacent angles whose sum is equal to two right angles.
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Euclid Book 1 Proposition 14
If at a point in a line, two other lines on opposite sides make the adjacent angles together equal to two right angles, then these two lines form one continuous line.
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Euclid Book 1 Proposition 15
If two lines intersect one another, the opposite angles are equal.
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Euclid Book 1 Proposition 16
If any side of a triangle is produced, then the exterior angle is bigger than either of the interior non-adjacent angles.
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Euclid Book 1 Proposition 17
The sum of any two angles of a triangle is less than two right angles.
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Euclid Book 1 Proposition 18
If in any triangle one side is longer than another, then the angle opposite to the longer side is bigger than the angle opposite to the shorter side.
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Euclid Book 1 Proposition 19
If in any triangle one angle is bigger than another, then the side opposite to the bigger angle is longer than the side opposite to the smaller angle.
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Euclid Book 1 Proposition 20
The sum of any two sides of a triangle is greater than the third.
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Euclid Book 1 Proposition 21
If two lines are drawn to a point within a triangle from the endpoints of its base, then their sum is less than the sum of the remaining sides, but they contain a bigger angle.
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Euclid Book 1 Proposition 22
To construct a triangle out of three line segments which equal three given line segments, the sum of every two of which is greater than the third.
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Euclid Book 1 Proposition 23
To construct an angle equal to a given angle on a given line segment and at a point on it.
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Euclid Book 1 Proposition 24
If two triangles have two sides of one respectively equal to two sides of the other, but the contained angle of one bigger than the contained angle of the other, then the base of that which has the bigger angle is longer than the base of the other.
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Euclid Book 1 Proposition 25
If two triangles have two sides of one respectively equal to two sides of the other, but the base of one longer than the base of the other, then the angle contained by the sides of that which has the longer base is bigger than the angle contained by the sides of the other.
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Euclid Book 1 Proposition 26a
If two triangles have two angles of one equal respectively to two angles of the other, and the sides between the corresponding angles are equal, then the triangles are congruent.
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Euclid Book 1 Proposition 26b
If two triangles have two angles of one equal respectively to two angles of the other, and the sides opposite one pair of equal angles are equal, then the triangles are congruent.
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Euclid Book 1 Proposition 27
If a line intersecting two lines makes the alternate angles equal to each other, then the two lines are parallel.
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Euclid Book 1 Proposition 28a
If a line intersecting two lines makes the exterior angle equal to its corresponding interior angle, then the two lines are parallel.
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Euclid Book 1 Proposition 28b
If a line intersecting two lines makes the sum of two interior angles on the same side equal to the sum of two right angles, then the two lines are parallel.
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Euclid Book 1 Proposition 29
If a line intersects two parallel lines, then: 1) the alternate angles are equal to one another. 2) the exterior angles are equal to the corresponding interior angles. 3) the two interior angles on the same side sum to two right angles.
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Euclid Book 1 Proposition 30
If two lines are parallel to the same line, then they are parallel to one another.
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Euclid Book 1 Proposition 31
To draw a line through a given point parallel to a given line.
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Euclid Book 1 Proposition 32
If any side of a triangle is produced, then the external angle is equal to the sum of the two internal non-adjacent angles, and the sum of the three internal angles is equal to two right angles.
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Euclid Book 1 Proposition 33
The lines which join the adjacent endpoints of two equal and parallel lines are equal and parallel.
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Euclid Book 1 Proposition 34
The opposite sides and the opposite angles of a parallelogram are equal to one another, and either diagonal bisects the parallelogram.
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Euclid Book 1 Proposition 35
Two parallelograms on the same base and between the same parallels have the same area.
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Euclid Book 1 Proposition 36
Two parallelograms on equal bases and between the same parallels have the same area.
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Euclid Book 1 Proposition 37
Two triangles on the same base and between the same parallels have the same area.
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Euclid Book 1 Proposition 38
Two triangles on equal bases and between the same parallels have the same area.
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Euclid Book 1 Proposition 39
Two triangles with the same area on the same side of the same base are between the same parallels .
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Euclid Book 1 Proposition 40
Two triangles with the same area on equal bases which form parts of the same line, and on the same side of the line, are between the same parallels.
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Euclid Book 1 Proposition 41
If a parallelogram and a triangle are on the same base and between the same parallels, the area of the parallelogram is twice the area of the triangle.
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Euclid Book 1 Proposition 42
To construct a parallelogram equal to a given triangle, with one angle equal to a given angle.
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Euclid Book 1 Proposition 43
Two lines each parallel to one pair of parallel sides of a parallelogram, and passing through a point on a diagonal of the parallelogram, divide it into four parallelograms, of which the two through which the diagonal does not pass, and which are called the complements of the other two, have the same area.
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Euclid Book 1 Proposition 44
On a given line segment, construct a parallelogram that is equal to a given triangle, and has one of its angles equal to a given angle.
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Euclid Book 1 Proposition 45
To construct a parallelogram equal to a given polygon, and having an angle equal to a given angle.
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Euclid Book 1 Proposition 46
To construct a square on a given line segment.
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Euclid Book 1 Proposition 47
In a right triangle the area of the square on the hypotenuse is equal to the sum of the areas of the squares on the other two sides.
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Euclid Book 1 Proposition 48
If the area of the square on one side of a triangle is equal to the sum of the areas of the squares on the remaining sides, then the angle opposite to that side is a right angle.