In any triangle (ABC), the square on any side subtending an acute angle (C) is less than the sum of the squares on the sides containing that angle, by twice the rectangle (BC, CD) contained by either of them (BC) and the intercept (CD) between the acute angle and the foot of the perpendicular on it from the opposite angle.
In an obtuse-angled triangle (ABC), the square on the side (AB) subtending the obtuse angle exceeds the sum of squares on the sides (BC, CA) containing the obtuse angle, by twice the rectangle contained by either of them (BC), and its continuation (CD) to meet a perpendicular (AD) on it from the opposite angle.
To divide a given finite line (AB) into two segments (in H), so that the rectangle (AB. BH) contained by the whole line and one segment may be equal to the square on the other segment.
If a line (AB) be bisected (at C) and divided externally (at D), the sum of the squares on the segments (AD, DB) made by the external point is equal to twice the square on half the line, and twice the square on the segment between the points of section.
If a line (AB) be bisected (at C) and divided into two unequal parts (at D), the sum of the squares on the unequal parts (AD, DB) is double the sum of the squares on half the line (AC), and on the segment (CD) between the points of section.
If a line (AB) be divided into two parts (at C), the square on the sum the whole line (AB) and either segment (BC) is equal to four times the rectangle contained by the whole line (AB) and that segment, together with the square on the other segment (AC).
If a right line (AB) be divided into any two parts (at C), the sum of the squares on the whole line (AB) and either segment (CB) is equal to twice the rectangle (2AB.CB) contained by the whole line and that segment, together with the square on the other segment.
If a line (AB) be bisected (at C), and divided externally in any point (D), the rectangle (AD.BD) contained by the segments made by the external point, together with the square on half the line, is equal to the square on the segment between the middle point and the point of external division.
If a line (AB) be divided into two equal parts (at C), and also into two unequal parts (at D), the rectangle (AD.DB) contained by the unequal parts, together with the square on the part (CD) between the points of section, is equal to the square on half the line.
If a line (AB) be divided into any two parts (at C), the square on the whole line is equal to the sum of the squares on the parts (AC, CB), together with twice their rectangle.
If a line (AB) be divided into two segments (at C), the rectangle contained by the whole line and either segment (CB) is equal to the square on that segment together with the rectangle contained by the segments.
If a line (AB) be divided into any two parts (at C), the square on the whole line is equal to the sum of the rectangles contained by the whole and each of the segments (AC, CB).
If there be two lines (A, BC), one of which is divided into any number of parts (BD, DE, EC), the rectangle contained by the two lines (A, BC), is equal to the sum of the rectangles contained by the undivided line (A) and the several parts of the divided line.
If the square on one side (AB) of a triangle be equal to the sum of the squares on the remaining sides (AC, CB), the angle (C) opposite to that side is a right angle.
In a right-angled triangle (ABC) the square on the hypotenuse (AB) is equal to the sum of the squares on the other two sides (AC, BC).
On a given right line (AB) to describe a square.
To construct a parallelogram equal to a given rectilineal figure (ABCD), and having an angle equal to a given rectilineal angle (X).
To a given, right line (AB) to apply a parallelogram which shall be equal to a given triangle (C), and have one of its angles equal to a given angle (D).
In any parallelogram the complements of the parallelograms about the diameter equal one another.
To construct a parallelogram equal to a given triangle (ABC), and having an angle equal to a given angle (D).
If a parallelogram (ABCD) and a triangle (EBC) be on the same base (BC) and between the same parallels, the parallelogram is double of the triangle.
Equal triangles (ABC, DEF) on equal bases (BC, EF) which form parts of the same right line, and on the same side of the line, are between the same parallels.
Equal triangles (BAC, BDC) on the same base (BC) and on the same side of it are between the same parallels.
Two triangles on the equal bases and between the same parallels are equal.
Triangles (ABC, DBC) on the same base (BC) and between the same parallels (AD, BC) are equal.
Parallelograms (BD, FH) on equal bases (BC, FG) and between then same parallels are equal.
Parallelograms on the same base (BC) and between the same parallels are equal.
The opposite sides (AB, CD,; AC, BD) and the opposite angles (A, D; B, C) of a parallelogram are equal to one another, and either diagonal bisects the parallelogram.
The right lines (AC, BD) which join the adjacent extremities of two equal and parallel right lines (AB, CD) are equal and parallel.
If any side (AB) of a triangle (ABC) be produced (to D), the external angle (CBD) is equal to the sum of the two internal non-adjacent angles (A, C), and the sum of the three internal angles is equal to two right angles.
Through a given point (C) to draw a right line parallel to a given right line.
If two right lines (AB, CD) be parallel to the same right line (EF), they are parallel to one another.
If a right line (EF) intersect two parallel right lines (AB, CD), it makes:
If a right line (EF) intersect two parallel right lines (AB, CD) makes the exterior angle (EGB) equal to its corresponding interior angle (GHD), or makes two interior angles (BGH, GHD) on the same side equal to two right angles, the two right lines are parallel.
If a right line (EF) intersecting two right lines (AB, CD) makes the alternate angles (AEF, EFD) equal to each other, these lines are parallel.
If two triangles (ABC, DEF) have two angles B, C) of one equal respectively to two angles (E, F) of the other, and a side of one equal to a side similarly placed with respect to the equal angles of the other, the triangles are equal in every respect.
If two triangles (ABC, DEF) have two sides (AB, AC) of one respectively equal to two sides (DE, DF) of the other, but the base (BC) of one greater than the base (EF) of the other, the angle (A) contained by the sides of that which has the greater base is greater them the angle (D) contained by the sides of the other.
If two triangles (ABC, DEF) have two sides (AB, AC) of one respectively equal to two sides (DE, DF) of the other, but the contained angle (BAC) of one greater than the contained angle (EDF) of the other, the base of that which has the greater angle is greater than the base of the other.
At a given point (A) in a given right line (AB) to make an angle equal to a given rectilineal angle (DEF)
To construct a triangle which three sides shall be respectively equal to three given lines $(A, B, C)$, the sum of every two of which is greater than the third.
If two lines $(BD, CD)$ be drawn to a point $(D)$ within a triangle from the extremities of its base $(BD)$, their sum is less than the sum of the remaining sides $(BA, CA)$, but they contain a greater angle.
The sum of any two sides $(BA, AC)$ of a triangle $(\triangle ABC)$ is greater than the third.
If one angle $(\angle B)$ of a triangle $(\triangle ABC)$ be greater than another angle $(\angle C)$, the side (AC) which it opposite to the greater angle is greater than the side $(AB)$ which is opposite to the less.
If in any triangle $(\triangle ABC)$ one side $(AC)$ be greater than another $(AB)$, the angle opposite to the greater side is greater than the angle opposite to the less.
Any two angles $(\angle B, \angle C)$ of a triangle $(\triangle ABC)$ are together less than two right angles.
If any side $(BC)$ of a triangle $(\triangle ABC)$ be produced, the exterior angle $(\angle ACD)$ is greater than either of the interior non-adjacent angles.
If Two right lines $(AB, CD)$ intersect one another, the opposite angles are equal $(CEA=DEB$, and $BEC=AED)$
If at a point $(B)$ in a right line $(BA)$ two other right lines $(CB, BD)$ on opposite sides make the adjacent angles $(\angle CBA, \angle ABD)$ together equal to two right angles, these two right lines form one continuous line.
The adjacent angles $(\angle ABC, \angle ABD)$ which one right lines $(AB)$ standing on another $(CD)$ makes with it are either both right angles, or their sum is equal to two right angles.
To draw a perpendicular to a given indefinite right line $(AB)$ from a given point $(C)$ without it.
From a given point $(C)$ in a given right line $(AB)$ to draw a right line perpendicular to the given line.
To bisect a given finite right line $(AB)$.
To bisect a given rectilineal angle $(\angle BAC)$.
If two triangles $(\triangle ABC, \triangle DEF)$ have two sides $(AB, AC)$ of one respectively equal to two sides $(DE, DF)$ of the other, and have also the base $(BC)$ of one equal to the base $(EF)$ of the other; then the two triangles shall be respectively equal to the angles of the – namely, those shall be equal to which the equal sides are opposite.
If two triangles $(\triangle ACB, \triangle ADB)$ on the same base $(AB)$ and on the same side of it have one pair of conterminous sides $(AC, AD)$ equal to one another, the other pair of conterminous sides $(BC, BD)$ must be unequal.
If two angles $(\angle B, \angle C)$ of a triangle be equal, the sides $(AC, AB)$ opposite to them are also equal.
The angles $(\angle ABC, \angle ACB)$ at the base $(BC)$ of an isosceles triangle are equal to one another, and if the equal sides $(AB, AC)$ be produced, the external angles $(\angle DBC, \angle ECB)$ below the base shall be equal.
If two triangle $(\triangle BAC, \triangle EDF)$ have two sides $(BA, AC)$ of one equal respectively to two sides
From the greater $(AB)$ of two given right lines to cut off a part equal to $(C)$ the less.
From a given point $(A)$ to draw a right line equal to a given finite right line $(BC)$.
On a given finite right line $(AB)$ to construct an equilateral triangl.
