THE BARYCENTRIC COORDINATES OF THE CENTRE OF AN ELLIPSE

The barycentric coordinates of the centre of an ellipse inscribed in a triangle

The barycentric coordinates of the centre of an ellipse inscribed in a triangle

Let Q be the centre and F, F΄ be the foci of an ellipse inscribed in the triangle ABC and tangent to the sides BC, CA, AB at the points K, L, M respectively.

We know that AK, BL, CM are cevians passing throught the same point P and let

(x, y, z ) be the barycentric coordinates of P.

If P is e.g the complex number orresponding to P then

P = , K = , L = , M =

It is known that the projections of the foci of the ellipse on the sides of the triangle lie on the primitive circle of the ellipse, that is on the pedal circle of the points F, F΄ and hence the points F, F΄ are isogonal conjugates.

It is also known that AF bisects the angle LFM and hence

 the number is real

or the number THE BARYCENTRIC COORDINATES OF THE CENTRE OF AN ELLIPSE is real

or the number is real

or the number is real.

Similarly we have that the numbers

must be real and therefore

xy(A  F)(B  F) + yz(B  F)(C  F) +zx(C  F)(A  F) = 0 (1)

because otherwise two of the numbers (A  F)², (B  F)², (C  F)² would have the same argument or two of the numbers A  F, B  F, C  F would have arguments that differ by 180^o which would mean that the point F lies on one of the sides of the triangle ABC, which is impossible.

From (1) we conclude that the foci of the ellipse are the roots of the equation

(xy+yz+zx)F²  [(xy+xz)A+(yz+yx)B+(zx+zy)C]F + xyAB + yzBC + zxCA = 0

and since Q =

we have that the barycentric coordinates of the centre Q are

x(y+z), y(z+x), z(x+y)

and from 2Q = 3 THE BARYCENTRIC COORDINATES OF THE CENTRE OF AN ELLIPSE = 3G  P^¹, we have

G = which means that Q is the complement of P^¹ where G is the centroid of ABC and P^¹ is the isotomic conjugate of P.

The Q cross conjugate of G is the point with barycentric coordinates

THE BARYCENTRIC COORDINATES OF THE CENTRE OF AN ELLIPSE

or x, y, z which is the point P.

So P is the Q cross conjugate of G. If we call Q as the elliptic centre of P then from Cimberling's encyclopaedia, we find that there are mentioned there, the elliptic centres of only 33 points, if we omit the trivial case of P = X(7) = Gergonne point with elliptic centre the point Q = X(1) = Incentre, where the ellipse is the incircle.

These points and the corresponding elliptic centre are the following.

point centre point centre point centre point centre

X(1) X(37) X(14) X(395) X(95) X(140) X(306) X(440)

X(2) X(2) X(66) X(32) X(98) X(230) X(315) X(206)

X(3) X(216) X(67) X(187) X(99) X(523) X(325) X(114)

X(4) X(6) X(69) X(3) X(189) X(57) X(329) X(223)

X(5) X(233) X(75) X(10) X(190) X(514) X(330) X(75)

X(6) X(39) X(76) X(141) X(253) X(4) X(508) X(178)

X(7) X(1) X(80) X(44) X(264) X(5) X(523) X(115)

X(8) X(9) X(85) X(142) X(280) X(281)

X(13) X(396) X(92) X(226) X(290) X(511)

and of these ellipses only one has its foci as points mentioned in Cimberling's encyclopaedia. Is the ellipse of P = X(264) = Isotomic conjugate of circumcentre

with elliptic centre the point X(5) = the ninepoint centre and foci at the points

X(3) = Circumcentre and X(4) = Orthocentre. 

Tags: barycentric coordinates, with barycentric, centre, coordinates, ellipse, barycentric