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III. 26, cor. 3.

III. 26, cor. 5.

If chords intersect at the same angle, within a circle,

And if they intersect without the circle,

But if one pair intersect within, and the other without the circle.

III. 26, cor. 4. . If two chords intersect within a circle at right angles. If a chord of a circle be produced till the produced part is equal to the radius, and if a line be drawn from its extremity through the center of the circle to meet the concave circumference. If, in equal circles, or the same circle, straight lines are equal.

III. 28.

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CONSEQUENCES.

They are equal to one another, and have equal arcs. They are equally distant from the center.

They are equal to one another.

That line is the greatest, And of all others, that which is nearer to the center is greater than the more remote,

And the greater is nearer to the center than the less. They intercept equal arcs. The angle formed by them is equal to the angle terminated at the circumference by the sum or difference of the arcs which they intercept, according as the point in which they meet is within or without the circle.

The sums of the arcs which they respectively intercept are equal;

The differences are equal;

The sum of the one pair of arcs

is equal to the difference of the other.

The sums of the opposite arcs intersected are equal.

The concave portion of the circumference intercepted is equal to three times the

convex.

They cut off equal parts of the circumferences, the greater equal to the greater, and the less to the less.

They are subtended by equal straight lines.

They intercept arcs the chords of which are parallel.

III. 22.

III. 22, cor. 1.

III. 36, cor. 2.

R. On Figures contained in Circles.

HYPOTHESES.

If a four-sided figure is con-
tained within a circle.

If one side of a four-sided
figure contained within a
circle be produced.

III. 22. schol. 2. If a four-sided figure has its
opposite angles together
equal to two right angles.
If the rectangle under the
segments, made by the in-
tersection of the diagonals
of a four-sided figure, are
equal in area,
Or if the rectangles under the
segments, made by pro-
ducing its opposite sides
to intersect, are equal in

area.

CONSEQUENCES.

Its opposite angles are to gether equal to two right angles.

The external angle is equal to the angle opposite to the internal adjacent angle.

A circle may be described about it.

The four-sided figure may have a circle described about it.

I. 2.

I. 31.

I. 3.

I. 10.
II. 11.

I. 34, schol.

I. 23.

I. 9.
I. 11.

I. 12.

III. 31, cor. 3.

I. 32 B, cor. 6.

PROBLEMS.

A. Relating to straight Lines.

From a given point to draw a straight line equal to a given finite straight line.

Through a given point to draw a straight line parallel to a given straight line.

From the greater of two given straight lines to cut off a part equal to the less.

To bisect a given finite straight line.

To divide a given finite straight line into two parts, so that the rectangle under the whole line and one segment shall be equal in area to the square on the other segment.

To divide a given finite straight line into any given number of equal parts.

B. Relating to rectilineal Angles.

At a given point in a given straight line to form a rectilineal angle equal to a given rectilineal angle.

To bisect a given rectilineal angle.

From a given point in a given straight line to draw a perpendicular to that line.

To draw a straight line perpendicular to a given straight line of an unlimited length, from a given point without it. To draw a straight line through the extremity of a given straight line, perpendicular to the same.

To bisect a given right angle.

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C. Relating to Triangles.

Given three finite straight lines, of which any two together
are greater than the third, to construct a triangle whose
sides shall be respectively equal to the given lines.

To construct an equilateral triangle upon a given finite
straight line.

Any two sides of a right-angled triangle being given, to
find the third side.

D. Relating to Parallelograms.

To construct a parallelogram equal in area to a giver
triangle, and having an angle equal to a given rectilinea
angle.

Upon a given finite straight line to construct a parallelo-
gram equal in area to a given triangle, and having an
angle equal to a given rectilineal angle.

To construct a parallelogram equal in area to a given
rectilineal figure, and having an angle equal to a giver
rectilineal angle.

Upon a given finite straight line to construct a parallelo
gram equal in area to a given rectilineal figure, and
having an angle equal to a given rectilineal angle.

To construct a rectangle under two given finite straight

lines.

Upon a given finite straight line to construct a square.
To construct a square equal in area to the sum of two or
more given squares.

To construct a square equal in area to the difference of two
given squares.

To construct a square equal in area to a given rectilineal
figure.

To find geometrical values of ✔ 1, √2, √3, &c.

E. Relating to Circles.

To find the center of a given circle.

From a given point, either without a given circle or in its
circumference, to draw a straight line touching the cir
cumference.

To draw a tangent to a given circle, from a given point
without it.

To bisect a given arc.

A segment of a circle being given, to describe the circle of
which it is a segment.

On a given finite straight line to describe a segment of
circle, which shall contain an angle equal to a giver
rectilineal angle.

To cut off from a given circle a segment which shall con
tain an angle equal to a given rectilineal angle.

3

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