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Geometrical Treatmentof Curves

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Geometrical Treatmentof Curves ‘ GEOMETRICAL TREATMENT OF CURVES W H IC H A R E ISOGONAL CONJUGATE TO A STRAIGHT LINE WITH RESPEC T TO A TRIANGLE IN T WO PA R TS PA R T FIR S T S A T T P n . D I . H J. C W , . UNIV E RS IT Y O F P E NNS Y L V A NIA II SH EWE LL AND SANBORN L EAC , , BO STO N NE W YO RK C HIC AGO CONTENTS O F PART FIRST . Tm: HYPE R B O LA Tm: EL LIPS E Second Pa rt cont a in i T a n E LLIPS E ( cont nu ed ) . Tm: PA R A B O LA . H IGH E R PL A NE CUR V E S . G OM TRICAL ATMEN F E E TRE T O CURVES. Two a are said be a con u ate h 1 . str ight lines to isogon l j g wit respect to an angle when they form equal angles with the Le n bisector of the angle . t the poi t of intersection of three a ac a an a a a an str ight lines, e h dr wn from ngul r point of tri gle, a a a be 0 , then their isogon l conjug tes will lso intersect in one sa a a a o point, y which is c lled the isogon l conjug te p int to a a . the point 0 , with respect to the three ngles of the tri ngle The isogona l conjugate points to all points on a straight line with respect to a triangle will form a curve which is said to be isogonal conjugate to the straight line with respect to the tri ' If l a a L Fi . r a . al s L 1 a e ngle the point on str ight line , g , ’ n a u a B and C a AB joi ed with the ng l r points , , of the tri ngle C, then we get two proj ective ! at pencils which are also perspec a tive to one nother, since the intersection of the corresponding The na i all rays are collinear . isogo l conjugate po nts to points ' ’ LL if B and C will ! at on , joined to , form two projective The a are pencils . intersections of two corresponding r ys in a a and are general not on the s me str ight line, therefore not and h u s a as w perspective ; t eir loc is conic section, is kno n l a r from the fundamenta l ws of projective geomet y . a u a a a 2. The conic isogon l conj g te to str ight line with respect to a triangle will pass through the angular points of ’ a B and C Fi . 1 the triangle . It will p ss through , g , since they in ou r !at . For are case the vertices of the pencils , let the point of intersection of the given line LL ' with the side B C be 2 I S OGONAL CUR VE S . P BA a u a B P and CA na a , then is isogon l conj g te to , ; is isogo l i u a CP . n B A and CA s conj g te to , The point of intersectio of a a the o BP the isogon l conjug te point to p int of intersection of , and CP A a a and , ; or, is the point isogon l conjug te to P,” the n i e r Ta and A as A . A C co ic w ll, ther fore, p s th ough king , or and B as !at ha B a , the vertices of the pencils, we ve the isogon l u a i P and a u a i conj g te po nt to , , C the isogon l conj g te po nt to P and n a a h u B and C. , ; the co ic p sses lso t ro gh The a ac a a 3. ch r ter of the conic isogon l conjug te to the line ' LL with respect to the triangle AB C depends u pon the posi tion of the line with respect to the circumcircle of the triangle AB C. We know that the point isogonal conjugate to a point In on the circu mcircle of the triangle is a point at infinity . order to find the isogonal conj u gate point to P on the circum of AB a e s circle C, we h ve to find the inters ction of the line ' Fi i re AA a u a e PA PB P . a isogon l conj g t to , , C ( g wh ch , ' ' B B , and CC . ' ' AA a a AP a BAA u a If is isogon l conjug te to , then ngle eq ls ' e B P BB angle PA C w) . L t the isogonal conj u gate to be ' P = x an le . B then g PBA equals angle CBB Since angle C , ' an le PBA al a BB as therefore g equ s ngle C B , denoting the I B 'BA a B . t ha a whole ngle by B follows, therefore, t t ngle ' ' I w ma u a as and BB i a a to AA . n am a eq ls , s p r llel the s e y it y ' ' ' ' to AA BB and n a a BB . be show that CC is p r llel Thus, , , ’ CC are ll ra ac and a u a a pa llel to e h other, the isogon l conj g te n P at i at n . poi t to , their intersect on, is therefore infi ity ' 1 e u m L L Fi . 4. If the line , g , do s not cut the circ circle of a i a a the tri ngle, then the con c isogon l conjug te to this line with e an h at i and resp ct to the tri gle as no point nfinity, the conic n s a will be a ellipse . If the line intersect the circle bout the a u a w a triangle, the conic isogon l conj g te to the line ill h ve two at t and an a . If points infini y, will, therefore, be hyperbol the m an h line touches the circu circle of the tri gle, t en the conic will a at fin and w e be a a a la . h ve one point in ity, ill , ther fore, p r bo 3 THE H YP E RB OLA . 1 . THE HYPE RBOLA . The n na a a t the 5. co ic isogo l conjug te to line which cu s a r a . to r a n circumcircle of t i ngle is, with respect this t i ngle, a ’ Kie ert s H a a a . hyperbol , which is c lled p yperbol m a u P and Fi . 3 If the str ight line cuts the circ circle in Q, g , ' will AP a a e AP m then the line , isogon l conjug t to , deter ine at i. e. di the direction of the point infinity , the rection of one of ' as o e a. The i A a con the ympt t s of the hyperbol l ne Q , isogon l a A di e m jug te to Q , will determine the rection of the s cond asy p The as a a tote . ( ymptotes will be p r llel to these directions . ) P and A a a as A Q form the s me ngle their isogonal conj u gates . ' S an B AP a a P A C and an A ince gle equ ls ngle , gle Q C equals ' ' a BA BAP CAB A C a P A ngle Q , therefore 4 + X + 4Q equ ls X C ' ' E BA an AF a a P ’ OA A . S K X Q , or gle Q equ ls ngle Q ince the asymptotes of the hyperbola isogonal conjugate to P Q with ' ' to a AB C are a a AP and A respect the tri ngle p r llel to Q , there fore the angle between the asympto tes of the hyperbola will be equ al to the angle between AP and A Q . If the line P Q a r u i n p sses th o gh the centre of the circle, then w ll the a gle formed by A P and A Q be a right angle ; the asymptotes of a a a to er en the hyperbol will be p r llel these lines, hence p p dicu lar ac and a i u i to e h other ; the hyperbol w ll be eq lateral . We ha am o a a n u 6. s ll now ex ine the hyperb l isogon l co j gate ’ ’ to Brocard s Diameter (produced) . Brocard s diameter is the ’ line joining the centre of the circu mcircle and Grebe s point of ’ has the triangle . Grebe s point the property that its distances from the sides of the triangle are to each other as the respec and a tive sides ; the sum of the squ res of these is , for the tri ’ A B a am e a a . s a ngle, minimum roc rd s di et r p sses through the r rc a n u cent e of the ci umcircle, the conic isogon l co j gate to it is an equ ilateral hyperbola .
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