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

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Page 18

When two lines L and M

' point of intersection , any one of which may be taken to be the angle between

the lines . These four angles consist of two pairs of opposite or vertical angles ,

viz ...

When two lines L and M

**cut**one another four angles are formed about the BA AB' point of intersection , any one of which may be taken to be the angle between

the lines . These four angles consist of two pairs of opposite or vertical angles ,

viz ...

Page 27

If P is not on the right bisector of AB , let the right bisector

QB , ( 53 ) but PA = PB , ( hyp . ) QP = PB - QB , PB = QP + QB , which is not true .

( 25 , Ax . ) Therefore the right bisector of AB does not

If P is not on the right bisector of AB , let the right bisector

**cut**AP in Q. Then QA =QB , ( 53 ) but PA = PB , ( hyp . ) QP = PB - QB , PB = QP + QB , which is not true .

( 25 , Ax . ) Therefore the right bisector of AB does not

**cut**AP ; and similarly it ... Page 28

( 53 ) If P is not on the right bisector , let AP B

= QB , and ZQAB = LQBA . ( 53 ° and Cor . 1 ) But LPBA = LQAB ; ( hyp . ) LPBA =

LQBA , which is not true unless P and Q coincide . Therefore if P is not on the ...

( 53 ) If P is not on the right bisector , let AP B

**cut**the right bisector in Q. Then QA= QB , and ZQAB = LQBA . ( 53 ° and Cor . 1 ) But LPBA = LQAB ; ( hyp . ) LPBA =

LQBA , which is not true unless P and Q coincide . Therefore if P is not on the ...

Page 43

... when it is called à right - angled triangle . All other triangles are called acute -

angled triangles , and have three acute angles . 6. The acute angles in a right -

angled triangle are complementary to one another . 78 ° . Theorem . - If a line

... when it is called à right - angled triangle . All other triangles are called acute -

angled triangles , and have three acute angles . 6. The acute angles in a right -

angled triangle are complementary to one another . 78 ° . Theorem . - If a line

**cuts**... Page 44

If a line

let N be any parallel to M. Then L

N. But M is 1l to N. Therefore through the same point P two lines L and M pass ...

If a line

**cuts**a given line it**cuts**every parallel to the given line . Let L**cut**M , andlet N be any parallel to M. Then L

**cuts**N. M L N Proof .-- If L does not**cut**N it is | toN. But M is 1l to N. Therefore through the same point P two lines L and M pass ...

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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.