## ELEMENTARY SYNTHETIC GEOMETRY OF THE POINT, LINE AND CIRCLE IN THE PLANE |

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Page 74

The sum of the alternate angles of any concyclic

) right angles . 5. If the angle of a trammel is 60 ° what arc of a circle will it

describe ? what if its angle is no ? 6. Trisect a right angle and thence show how

to draw ...

The sum of the alternate angles of any concyclic

**polygon**of 2n sides is 2 ( 12 – 1) right angles . 5. If the angle of a trammel is 60 ° what arc of a circle will it

describe ? what if its angle is no ? 6. Trisect a right angle and thence show how

to draw ...

Page 86

REGULAR

entrant angles ( 89 ° , 2 ) is in general called a

according to the number of their sides as follows : 3 , triangle or trigon ; 4 ,

quadrangle , or ...

REGULAR

**POLYGONS**. 132o . Def . 1. -A closed rectilinear figure without re -entrant angles ( 89 ° , 2 ) is in general called a

**polygon**. They are namedaccording to the number of their sides as follows : 3 , triangle or trigon ; 4 ,

quadrangle , or ...

Page 87

regular

Proof - Let AB , BC be two consecutive sides of the

Then the triangles AOB , BOC are isosceles and congruent . LOAB = LOBA --

LOBC ...

regular

**polygon**, the magnitude of an internal angle is ( 2 - 4 ) right angles . A B сProof - Let AB , BC be two consecutive sides of the

**polygon**and O its centre .Then the triangles AOB , BOC are isosceles and congruent . LOAB = LOBA --

LOBC ...

Page 88

Problem .--- To determine which species of regular

can fill the plane . That a regular

the plane , the number of right angles in its internal angle must be a divisor of 4.

Problem .--- To determine which species of regular

**polygons**, each taken alone ,can fill the plane . That a regular

**polygon**of any species may be capable of fillingthe plane , the number of right angles in its internal angle must be a divisor of 4.

Page 98

To construct a triangle equal to a given

drawn through a given point in one of the sides . 4. To construct a rhombus equal

to a given parallelogram , and with one of the sides of the parallelogram as its ...

To construct a triangle equal to a given

**polygon**. 3. To bisect a triangle by a linedrawn through a given point in one of the sides . 4. To construct a rhombus equal

to a given parallelogram , and with one of the sides of the parallelogram as its ...

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### Common terms and phrases

ABCD algebraic altitude base becomes called centre centre-line chord circle coincide collinear common concurrent congruent considered constant construct Converse corresponding cuts denote describe determine diagonals diameter difference direction distance divided draw drawn equal expressed external figure fixed point four geometric given given line given point gives greater harmonic Hence internal intersect inverse joins length lies line-segment mean measure median meet middle point opposite opposite sides orthogonally pair parallel passes pencil perpendicular perspective placed plane polar polygon position Proof proved quadrangle radical axis radius range ratio reciprocal rectangle regular relation remaining respect right angle right bisector rotation segment sides similar Similarly similitude square straight symbol taken tangent theorem touch triangle vertex vertices

### Popular passages

Page 176 - The square described on the hypothenuse of a rightangled triangle is equal to the sum of the squares described on the other two sides.

Page 183 - The sides of a triangle are proportional to the sines of the opposite angles.

Page 260 - Three lines are in harmonical proportion, when the first is to the third, as the difference between the first and second, is to the difference between the second and third ; and the second is called a harmonic mean between the first and third. The expression 'harmonical proportion...

Page 19 - Every circumference of a. circle, whether the circle be large or small, is supposed to be divided into 360 equal parts called degrees. Each degree is divided into 60 equal parts called minutes, and each minute into 60 equal parts called seconds.

Page 77 - J_ to the given line. .'. the construction gives the perpendicular to a given line at a given point in the line.

Page 122 - And conversely, if the square on one side of a triangle is equal to the Bum of the squares on the other two sides, the angle contained by these two sides is a right angle.

Page 124 - The difference of the squares of two sides of a triangle is equal to the difference of the squares of the segments made by the altitude upon the third side.