## An Elementary Treatise on Trilinear Co-ordinates: The Method of Reciprocal Polars, and the Theory Projections |

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Page 2

1 . : Similarly bB = twice the area of PCA , cy = twice the area of PAB . Adding

these equations , we get aa + b3 + cy = 2A . Next , suppose P to lie between AB ,

AC

2 .

1 . : Similarly bB = twice the area of PCA , cy = twice the area of PAB . Adding

these equations , we get aa + b3 + cy = 2A . Next , suppose P to lie between AB ,

AC

**produced**, and on the side of BC remote from A ( fig . 2 ) . Then a will be Fig .2 .

Page 3

Hence , twice the area PBC will be represented by — ad , and we shall therefore

have as before aa + b3 + cy = 2A . Thirdly , let P lie between AB , AC ,

backwards ( fig . 3 ) , so that B , y are negative while a is positive . Twice Fig . 3 .

Hence , twice the area PBC will be represented by — ad , and we shall therefore

have as before aa + b3 + cy = 2A . Thirdly , let P lie between AB , AC ,

**produced**backwards ( fig . 3 ) , so that B , y are negative while a is positive . Twice Fig . 3 .

Page 22

... joining the centre of the circle inscribed in the triangle ABC , with the middle

point of the side BC , is parallel to the straight line joining A with the point of

contact of the circle touching BC externally and AB , AC

sides BC ...

... joining the centre of the circle inscribed in the triangle ABC , with the middle

point of the side BC , is parallel to the straight line joining A with the point of

contact of the circle touching BC externally and AB , AC

**produced**. 6 . On thesides BC ...

Page 26

The Method of Reciprocal Polars, and the Theory Projections Norman Macleod

Ferrers. Fig . 10 . P Р R Let the angle POR be bisected internally by OQ , let PO

be

...

The Method of Reciprocal Polars, and the Theory Projections Norman Macleod

Ferrers. Fig . 10 . P Р R Let the angle POR be bisected internally by OQ , let PO

be

**produced**to any point P ' , and let the angle P OR be bisected by OS , then sin...

Page 28

Join B'C , BC " , and

fourth harmonic required . For , let ABC be the triangle of reference , and let the

equation of AD be ß - kry = 0 . " Let the equation of B C be na + ß - koy = 0 .

Join B'C , BC " , and

**produce**them to meet in E. Join AE , then AE shall be thefourth harmonic required . For , let ABC be the triangle of reference , and let the

equation of AD be ß - kry = 0 . " Let the equation of B C be na + ß - koy = 0 .

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### Common terms and phrases

angle angular points Arithmetic asymptotes auxiliary become BOOKS called Cambridge centre Chap chapter chord circle cloth co-ordinates coincide College condition conic conic section considered constant corresponding Crown 8vo curve denoted described determine directrix distance equal equation Examples expressed fixed point focus follows four points given conic given point given straight line gives harmonic Hence hyperbola imaginary line at infinity locus meet move obtain opposite sides pair parabola parallel passing pencil perpendicular plane point of intersection points of contact polar pole positive produced projection proposition prove ratio reciprocal relation represented respect result right angles satisfy Schools second degree seen sides similar Similarly suppose taken tangents term theorem third three points tion touch Treatise triangle of reference values whence writing written