RELFE BROTHERS' EUCLID SHEETS.
PROPOSITIONS 1–26, BOOK I, ARE NOW PUBLISHED IN A SIMILAR FORM TO THIS.

PROPOSITION XI.

To draw a straight line at right angles to a given straight line, from a given point in the same.

Let be the given straight line, and a given point in it.
It is required to draw a straight line from the point at right angles to

In take any point

and make
equal to

; upon

describe the equilateral triangle and join

Then drawn from the point shall be at right angles to Because is equal to and is common to the two triangles

; the two sides are equal to the two sides

each to each ; and the base is equal to the base ; therefore the angle is equal to the angle : and these two angles are adjacent angles. But when the two adjacent angles which one straight line makes with another straight line, are equal to one another, each of them is called a right angle: therefore each of the angles is a right angle. Wherefore from the given point in the given straight line

has been drawn at right angles to

Cor. By help of this problem, it may be demonstrated that two straight lines cannot have a common segment. If it be possible, let the segment be common to the two straight lines

.

From the point, draw at right angles to ; then because

is a straight line, therefore the angle is equal to the angle

Similarly, because is a straight line, therefore the angle is equal to the angle ; but the angle is equal to the angle wherefore the angle is equal to the angle the less equal to the greater angle, which is impossible.

Therefore two straight lines cannot have a common segment.

RELFE BROTHERS' EUCLID SHEETS.

PROPOSITIONS 1–26, BOOK I, ARE NOW PUBLISHED IN A SIMILAR FORM TO THIS.

PROPOSITION X.

To bisect a given finite straight line, that is, to divide it into two equal parts.

shall be cut into two equal parts in the point

RELFE BROTHERS' EUCLID SHEETS.

PROPOSITIONS 1–26, BOOK I, ARE NOW PUBLISHED IN A SIMILAR FORM TO THIS.

PROPOSITION IX.

To bisect a given rectilineal angle, that is, to divide it into two equal angles.