RELFE BROTHERS' EUCLID SHEETS.

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

PROPOSITION XXI.

If from the ends of a side of a triangle, there be drawn two straight lines to a point within the triangle ; these shall be less than the other two sides of the triangle, but shall contain a greater angle.

Let be a triangle, and from the points , the ends of the side let the two straight lines

be drawn to a point within the triangle. Then and shall be less than and the other two sides of the triangle, but shall contain an angle greater than the angle

Produce

to meet the side

in

Because two sides of a triangle are greater than the third side, therefore the two sides of the triangle

are greater than ; to each of these unequals add ; therefore the sides

are greater than Again, because the two sides

of the triangle
are greater than

; add to each of these unequals; therefore the sides

are greater than

But it has been shown that

are greater than

; much more then

greater than

are

Again, because the exterior angle of a triangle is greater than the interior and opposite angle; therefore the exterior angle

of the triangle

is greater than the interior and opposite angle ; for the same reason the exterior angle

of the triangle is greater than the interior and opposite angle ; and it has been demonstrated, that the angle is greater than the angle ; much more therefore is the angle

greater than the angle Therefore, if from the ends of the side, &c.

RELFE BROTHERS EUCLID SHEETS.

'

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

PROPOSITION XX.

Any two sides of a triangle are greater together than the third side.

Let

be a triangle.

Then any two sides of it together shall be greater than the third side, viz., the sides

and that the greater angle is subtended by the greater side ; therefore the side

is greater

than the side

; but

is equal to

and

therefore the sides

are greater than

; in the same manner it may be demonstrated, that the sides

are greater than

; also that

are greater than

Therefore any two sides, &c.

RELFE BROTHERS' EUCLID SHEETS.

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

PROPOSITION XIX.

The greater angle of every triangle is subtended by the greater side, or, has the greater side opposite to it.