## Elementary geometry |

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Side 140

... cut the circle , prove that the locus of · the middle points of the chords so formed

is a circle . 8 . In any inscribed hexagon the sum of any three alternate angles is

equal to four right angles . SECTION IV . A .

... cut the circle , prove that the locus of · the middle points of the chords so formed

is a circle . 8 . In any inscribed hexagon the sum of any three alternate angles is

equal to four right angles . SECTION IV . A .

**TANGENTS**( treated directly ) Sect . Side 142

A .

length which meets the circumference of a circle in two points . THEOREM 18 .

Every straiglet line through a point on the circumference of a circle meets it in one

...

A .

**TANGENTS**( treated directly ) . Def . 10 . A secant is a straight line of unlimitedlength which meets the circumference of a circle in two points . THEOREM 18 .

Every straiglet line through a point on the circumference of a circle meets it in one

...

Side 145

Two

Part En . Let A be a point external to the given circle BCD ; it is required to prove

that two , and only two , straight lines can be drawn from A to touch the circle BCD

.

Two

**tangents**, and only two , can be drawn to a circle from an external point .Part En . Let A be a point external to the given circle BCD ; it is required to prove

that two , and only two , straight lines can be drawn from A to touch the circle BCD

.

Side 146

For because the angle between the radius and the

. 18 . ) therefore the point of contact lies on the circle described on AO as

diameter . . ( 111 . 16 and 111 . 13 Obs . ) But this circle cannot intersect the given

circle ...

For because the angle between the radius and the

**tangent**is a right angle , ( 111. 18 . ) therefore the point of contact lies on the circle described on AO as

diameter . . ( 111 . 16 and 111 . 13 Obs . ) But this circle cannot intersect the given

circle ...

Side 147

Therefore there cannot be drawn more than two

COR . The two

make equal angles with the straight line joining that point with the centre . For let

AT ...

Therefore there cannot be drawn more than two

**tangents**from A to the circle .COR . The two

**tangents**drawn to a circle from an external point are equal andmake equal angles with the straight line joining that point with the centre . For let

AT ...

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ABCD adjacent angle ABC angle APB angle BAC base bisect bisector Book called centre chord circle circumference coincide College common Constr construct converse Crown 8vo diagonals diameter difference distance divided double draw drawn Edition ELEMENTARY equal angles Euclid Examples exterior angle extremities fall figure Find follows four Geometry given angle given line given point given straight line greater half Hence Illustrations inscribed interior intersect isosceles triangle Join length less Let ABC line drawn locus magnitudes measure meet middle point minor arc opposite sides parallel parallelogram pass perpendicular placed polygon PROBLEM produced Professor Proof proportional quadrilateral radius ratio rectangle contained rectilineal figure regular required to prove respectively right angles School segment Shew sides similar Similarly square supplementary tangent THEOREM third touch triangle ABC unequal

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