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

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Page 44

Thus

opposite vertices are the diagonals of the quadrangle . qe.cl. B A D 3. The

quadrangle formed when two parallel lines intersect two 44 SYNTHETIC

GEOMETRY .

Thus

**ABCD**is a quadrangle . 2. The line - segments AC and BD which joinopposite vertices are the diagonals of the quadrangle . qe.cl. B A D 3. The

quadrangle formed when two parallel lines intersect two 44 SYNTHETIC

GEOMETRY .

Page 51

BCD = 1 , ( 90 ° ) LBCD is supplementary to _BAD . But LBCD is supplementary

to LECD ; and the < ( BC , CD ) is either the angle BCD or DCE . ( 39 ° ) L ( BC.

**ABCD**is a quadrangle , and the Ls at B and D are right angles ; ( hyp . ) LBAD +BCD = 1 , ( 90 ° ) LBCD is supplementary to _BAD . But LBCD is supplementary

to LECD ; and the < ( BC , CD ) is either the angle BCD or DCE . ( 39 ° ) L ( BC.

Page 62

Now , let

D , or Dz . But in the latter case the normal quadra le

quadrangle ABCD2 , and the theorem remains true . Or , the theorem is ...

Now , let

**ABCD**be a quadrangle . Then the condition is not violated if D moves toD , or Dz . But in the latter case the normal quadra le

**ABCD**becomes the invertedquadrangle ABCD2 , and the theorem remains true . Or , the theorem is ...

Page 65

A quadrangle which has its opposite angles supplementary has its vertices

concyclic . ( Converse of 106 , Cor . 2 )

is supplementary to LABC ; then a circle can pass through A , B , C , and D. Proof.

A quadrangle which has its opposite angles supplementary has its vertices

concyclic . ( Converse of 106 , Cor . 2 )

**ABCD**is a quadrangle whereof the ZADCis supplementary to LABC ; then a circle can pass through A , B , C , and D. Proof.

Page 71

If four circles touch two and two externally , the points of contact are concyclic .

Let A , B , C , D be the centres of the circles , and P , Q , R , S be the points of

contact . Then AB passes through P , BC through Q , etc. ( 113 ° , Cor . 1 ) Now ,

If four circles touch two and two externally , the points of contact are concyclic .

Let A , B , C , D be the centres of the circles , and P , Q , R , S be the points of

contact . Then AB passes through P , BC through Q , etc. ( 113 ° , Cor . 1 ) Now ,

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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.

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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.