Elementary Synthetic Geometry of the Point, Line and Circle in the Plane |
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Page 3
... converse to one another when the hypothesis and conclusion of the one are respectively the conclusion and hypothesis of the other . Ex . If an animal is a horse it has four legs . Converse . If an animal has four legs it is a horse . As ...
... converse to one another when the hypothesis and conclusion of the one are respectively the conclusion and hypothesis of the other . Ex . If an animal is a horse it has four legs . Converse . If an animal has four legs it is a horse . As ...
Page 4
... converses of the follow- ing theorems : - I. The sum of two odd numbers is an even number . 2. A diameter is the longest chord in a circle . 3. Parallel lines never meet . 4. Every point equidistant from the end - points of a line ...
... converses of the follow- ing theorems : - I. The sum of two odd numbers is an even number . 2. A diameter is the longest chord in a circle . 3. Parallel lines never meet . 4. Every point equidistant from the end - points of a line ...
Page 26
... of the base of an isosceles tri- angle is the internal bisector of the vertical angle . And since these two bisectors are one and the same line the converse is true . Def . 2. -A triangle in which all the sides 26 SYNTHETIC GEOMETRY .
... of the base of an isosceles tri- angle is the internal bisector of the vertical angle . And since these two bisectors are one and the same line the converse is true . Def . 2. -A triangle in which all the sides 26 SYNTHETIC GEOMETRY .
Page 27
... ( Converse of 53 ° . ) PA = PB . Then P is on the right bisector of AB . Proof . If P is not on the right bisector of AB , let the right bisector cut AP in Q. Then but :: or which is not true . QA = QB , PA = PB , ( 53 ° ) ( hyp . ) QP ...
... ( Converse of 53 ° . ) PA = PB . Then P is on the right bisector of AB . Proof . If P is not on the right bisector of AB , let the right bisector cut AP in Q. Then but :: or which is not true . QA = QB , PA = PB , ( 53 ° ) ( hyp . ) QP ...
Page 28
... ( Converse of 53 ° , Cor . 1. ) :: Then But < PAB = 2PBA , then PA = PB . Proof . If P is on the right bisector of AB , PA = PB . ( 53 ° ) If P is not on the right bisector , let AP B cut the right bisector in Q. QA QB , and 4QAB = LQBA ...
... ( Converse of 53 ° , Cor . 1. ) :: Then But < PAB = 2PBA , then PA = PB . Proof . If P is on the right bisector of AB , PA = PB . ( 53 ° ) If P is not on the right bisector , let AP B cut the right bisector in Q. QA QB , and 4QAB = LQBA ...
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Common terms and phrases
ABCD Algebra altitude becomes bisects c.p.-circles centre of similitude chord of contact circles touch circumcircle co-axal coincide collinear concurrent concurrent lines concyclic congruent cut the circle denote diagonals diameter divided double points end-points equal angles equianharmonic equilateral triangle excircles external bisector fixed point geometric given circles given line given point harmonic range Hence hexagram homographic homologous hypothenuse incircle internal angles inverse points isosceles joins LAOB line-segment locus median middle point nine-points circle opposite sides orthogonally pair parallel parallelogram passes pencil perpendicular perspective plane point of contact point of intersection polar reciprocal Proof quadrangle radical axis radical centre radii radius rectangle rectilinear figure regular polygon rhombus right angle right bisector rotation secant similar Similarly square straight angle symbol tangent tensor theorem Theorem.-The three circles transversal vertex vertices