2. Conics with common perspector.

3. Harmonic ranges.

4. The centre of a circumellipse.

5. Common centre and asymptote.

6. Circumconics with common tangent.

7. Circumparabolas.

8. A result related to 4.

9. Some more circumellipses.

10. The centre and perspector of a circumconic.

11. The intersection of circumconics.

12. A sequence of intersection points.

13. Some centres of circumellipses.

14. An identity for the X(2)-Ceva conjugate.

15. The Steiner ellipses.

16. Locating the centres of circumellipses.

**1. Complements and anticomplements.**

The complement of a point P is the mid-point of P and its anticomplement.

The centroid G = [1,1,1] lies on the line through these points.

Further, (P,P^{-},P^{+},G) = -1, so we have a harmonic range.

Given a point P = [p,q,r], the complement is P^{-} = [q+r,r+p,p+q],

the anticomplement is P^{+} = [(q+r-p),(p+r-q),(p+q-r)].

If P is normalized, so that p+q+r=1, then so is P^{+}.

Then the mid-point is just the "barycentric sum", so is P^{-}.

The line PP^{-} has equation (q-r)x+(r-p)y+(p-q)z = 0.

This clearly contains P^{+} and G.

Now P^{+} = -P + P^{-}, G = P + P^{-}, so we get the cross-ratio.

Note that we also have (G,P^{+},P^{-},P) = -1.

**2. Conics with common perspector.**

If an inconic **I** and a circumconic **C** have common perspector P,

then
P and the centres, I, C, of the conics are collinear.

The barycentric square P^{2} also lies on the line of centres.

Further (P,C,I,P^{2}) = -1, so we have a harmonic range.

Say P = [p,q,r]. Then the centre of **I** is I = [p(q+r),q(r+p),r(p+q)].

The centre of **C** is C = [p(q+r-p),q(r+p-q),r(p+q-r)].

The line PI has equation x(q-r)/p+y(r-p)/q+z(p-q) = 0.

Then it is trivial to check that C and P^{2} = [p^{2},q^{2},r^{2}] lie on PI.

Let s = p+q+r, then P^{2} = (sP-C)/2, I = (sP+C)/2. This gives the

stated value for the cross-ratio.

If we use the barycentric product, then we can rewrite C = PP^{+},

I = PP^{-} = (P^{-1})^{-}.

**3. Harmonic ranges**

Since complementation and anticomplementation are projective,

they preserve cross-ratio and hence harmonic ranges.

If we fix a point K, then mapping P to KP is also projective,

so preserves collinearity and cross-ratio. We can view (2) above

as a special case of (1), with multiplier P.

**4. The centre of a circumellipse.**

If **C** is a circumellipse, with centre C and perspector P, then

C = [sin(U),sin(V),sin(W)], P = [sin^{2}(U/2),sin^{2}(V/2),sin^{2}(W/2)],

where U+V+W = 0.

A point of the form P with U+V+W = 2Kπ lies inside the Steiner

Inellipse, so is the
perspector of a circumellipse.

Thus **C** is a circumellipse if and only if the centre C has the form

[sin(U),sin(V),sin(W)] with U+V+W = Kπ.

Note. We may replace U, V, W by U+kπ, V+lπ, W+mπ with k,l,m

all odd or all even to get "angles" which sum to any multiple of 2π.

If the multipliers are odd, this reverses the sign of *all* of the

coordinates of C.

Examples are the circumcircle with U=2A,V=2B,W=2C, and the

Steiner
Ellipse with U=V=W=2π/3.

The proofs are non-trivial. you can find them here.

**5. Common centre and asymptote.**

Let **L** be the line xu+yv+zw = 0. Suppose that there are a

concentric inconic **I** and circumconic **C** with **L** as asymptote.

Then U = [u,v,w] lies on the Steiner Ellipse.

The other asymptote of **C** is **L ^{-1}** : x/u+y/v+z/w = 0.

The common centre is [(v-w)/u,(w-u)/v,(u-v)/w].

The perspector of

The perspector of

The other asymptote of

Although the result is lengthy, most of the proofs are very short.

We know that *the* circumconic with asymptote **L** has the stated

centre and perspector. Its other asymptote is **L ^{-1}**.

We also know that the inconic with asymptote

the circumconic having as asymptotes the lines xf+yg+zh = 0

and x/f+y/g+z/h = 0, with

[f,g,h] = [1/v+1/w-1/u,1/w+1/u-1/v,1/u+1/v-1/w].

One of these must be

u=v=w, so

If it is the latter, f = k/u, g = k/v, h = k/w for some k. Then either

k = 1, and u = v = w, which we saw is impossible, or we must have

1/u+1/v+1/w = 0. Thus U is on

Note that, as 1/u+1/v+1/w = 0, [f,g,h] = [1/u,1/v,1/w]. Then the

second asymptote of **I** is **L ^{2}**. The perspector is as stated.

Note. G = [1,1,1] lies on **L ^{-1}**.

**6. Circumconics with common tangent.**

Let **L** be the line ux+vy+wz = 0.

The circumconics touching **L** have perspectors on the inconic

with perspector U^{-1} = [1/u,1/v,1/w].

The circumconic which touches **L** at P = [p,q,r] has perspector

Q = [up^{2},vq^{2},wr^{2}]. **ref**

As P is on **L** up+vq+wr = 0. It is easy to check that Q lies on

the stated inconic.

Note. The inconic is the dual of the isotomic conjugate of **L**.

**7. Circumparabolas**

If the circumparabola **C** touches the line at infinity at U = [u,v,w],

then **C** has centre U^{2} = [u^{2},v^{2},w^{2}] which lies on the
Steiner Inellipse

and perspector U.

These results are just a special case of earlier results for circumconics. **ref**

The remark about the location of U^{2} follows from **6** since here the line

is x+y+z = 0, so the inconic has perspector [1,1,1]. This is the Steiner

Inellipse.

**8. A result related to 4.**

Suppose that u+v+w = π. Then the inconic **I** with centre

Q = [sin(u),sin(v),sin(w)] has perspector P' = [tan(u/2),tan(v/2),tan(w/2)].

We know that P' = (Q^{+})^{-1}. The result follows from the simple calculation

sin(v)+sin(w)-sin(u) = 4cos(u/2)sin(v/2)sin(w/2), and similar considerations

for the y- and z-coordinates.

In the notation of **4**, we make the choice of U,V,W so that U+V+W = 2π.

Then put 2u = U, 2v = V, 2w = W, so that the above result can be applied.

Note that the* centre* Q is a square root of the *perspector* P in **4**.

If the circumconic **C** of **4** is the circumcircle, then Q = [a,b,c]. Thus the

inconic **I** is the incircle, and the perspector is [tan(A/2),tan(B/2),tan(C/2)],

the Gergonne point.

If the circumconic **C** of **4** is the Steiner Ellipse, then Q = [1,1,1]. Thus the

inconic **I** is the Steiner Inellipse, and the perspector is again Q, the centroid.

**9. Some more circumellipses.**

We begin with an example :

The incentre I can be written as [sin(A+π),sin(B+π),sin(C+π)].

By **4**, the circumconic **C1** with centre I is an ellipse as the angle sum is 4π.

As in **4**, the perspector can be written as the barycentric square J^{2},

where J = [sin((A+π)/2,sin((B+π)/2),sin((C+π)/2)]. Now J has angle sum

equal to 2π, so is the centre of a further circumellipse **C2**. By **4 **this has

perspector K^{2}, where K = [sin((A+π)/4,sin((B+π)/4),sin((C+π)/4)].

This has angle sum π, so we add π to each angle to allow us to proceed.

Observe that the adjusted version of K has angle sum 4π, the same as I.

Thus we may proceed to define a sequence of circumellipses. At every

other stage, we need to add π to each angle.

Notes.

The perspector of **C1** is the Mittenpunkt, X(9) in ETC.

J = [cos(A/2),cos(B/2),cos(C/2)], so is the point X(188).

The perspector of **C2** may be found from **4**. It is X(236).

As we have seen this is the square of the centre of a further

circumellipse **C3**.

The incentre arises in this way from the circumcircle **C0**.

In turn, **C0** arises from the circumellipse with centre

X(1147) = [sin(4A),sin(4B),sin(4C)].

We now have a chain of four circumellipses with known centres.

Of course, we see that we can continue *backwards* along the

chain with centres [sin(2^{n}A),sin(2^{n}B),sin(2^{n}C)].

We have used nothing special about I except that it is the centre

of a circumellipse.

Suppose that **C** is a circumellipse with centre C and perspector P^{2}.

Then we can define a sequence of circumellipses {**C**(n)} as follows:

**C**(1) is **C**,

**C**(n+1) is the circumellipse with centre **C**(n+1) = P(n), where

P(n)^{2} is the perspector of **C**(n).

As n tends to infinity the **C**(n) tend to the Steiner Ellipse.

The example shows the heart of the argument. We begin by writing C in the

form [sin(U),sin(V),sin(W)], where U+V+W = 2kπ. We can choose U,V,W in

the range (0,2π), so that k is 1 or 2. The *detail* is slightly different in these

cases, but the reader should be able to make the necessary adjustments

for the case k = 1.

Suppose that k = 2. As in the example, we create the sequence.

We will calculate the first angle U(n) only. The others are similar.

Observe that, if n is even, then the angle sum is 4π, and then

U(n+1) = U(n)/2. Now the angle sum is 2π, so U(n+2) = U(n+1)/2+π

and the angle sum is 4π again.

If we concentrate on the even values of n, we see that we have

U(2m) = U/2^{2m} + π(1 + 1/4 + ... + 1/4^{m-1}).

As m tends to infinity, U(2m) tends to 4π/3.

Also U(2m+1) = U(2m)/2, so these tend to 2π/3.

Thus in either case, the centres tend to the centroid G = [1,1,1]

and the circumellipses to the Steiner Ellipse.

**10. The centre and perspector of a circumconic.**

The circumconic with perspector P = [p,q,r] has centre

C = [p(q+r-p),q(r+p-q),r(p+q-r)].

This is the X(2)-Ceva conjugate of P.

There are 21 examples of such pairs in ETC.

These include (X1),X(9)), (X(188),X(236)) we met in **9**.

Other examples are (X(2),X(2)), (X(3),X(6)), (X(4),X(1249))

and (X(5),X(216)).

**11. The intersection of circumconics.**

If circumconics have perspectors [p,q,r] and [p',q',r'], then

the fourth intersection is [1/(qr'-q'r),1/(rp'-r'p),1/(p'q-pq')].

This is just the observation that the conics are the isotomics of

the lines px+qy+rz = 0 and p'x+q'y+r'z = 0. The intersection

of the conics other than X,Y,Z is the isotomic conjugate of the

intersection of the lines.

If we take the circumcircle and the circumellipse with centre X(1),

the perspectors are X(6) and X(9), so the intersection is X(100).

For the circumcircle and the circumellipse with centre X(1147),

the fourth intersection is X(110).

**12. A sequence of intersection points.**

Suppose we have the sequence {**C**(n)} with **C**(1)a given ellipse.

If n > 1, say **C**(n) has centre [sin(U),sin(V),sin(W)]. Then the centre

of **C**(n-1) can be written as [sin(2U),sin(2V),sin(2W)]. By **11**,

the fourth intersection of these conics is the isotomic conjugate of

the point with x-coordinate sin(V)sin(2W) - sin(2V)sin(W). This can

be rewritten as 2sin(U)sin(V)sin(W)sin((V-W)/2)/sin(U/2). Now we

can remove the symmetrical factor to get sin((V-W)/2)/sin(U/2).

As n tends to infinity, U/2 approaches 2π/3 or π/3 which have the

same sine. Also, from **11**, (V-W) = (V*-W*)/2^{n}, where the centre

of **C**(1) has angles U*,V*,W*. If we consider the *ratios* of the

coordinates and use the result that sin(αx)/sin(βx) tends to α/β.

as x tends to zero, the fourth intersection tends to the point

[1/(V*-W*),1/(W*-U*),1/(U*-V*)].

When **C**(1) is the circumcircle, the limit point is [1/(B-C),1/(C-A),1/(A-B)].

**13. Some centres of circumellipses.**

We have seen that X(1), X(2), X(3), X(188) are the centres of circumellipses.

We can check that the following also qualify:

The Mittenpunkt X(9) = [a(b+c-a),b(c+a-b),c(a+b-c)], perspector X(1).

The Speiker centre X(10) =[b+c,c+a,a+b], perspector X(37), and

X(37) = [a(b+c),b(c+a),c(a+b)], perspector X(10).

In each case, we check that the corresponding perspector is inside the

Steiner Ellipse.

Cabri shows that X(4), X(5), X(6), X(7), X(8) do not always give ellipses.

**14. An identity for the X(2)-Ceva conjugate and others.**

A straight-forward calculation shows that the X(2)-Ceva conjugate can be

expressed as the complement of the isotomic conjugate of the anticomplement.

Thus, the X(2)-Ceva conjugate of U is ((U^{+})^{-1})^{-}.

This makes it clear why the transform has order 2.

In a similar way, the P-Ceva conjugate of U is constructed as follows

P (((U/P)^{+})^{-1})^{-}, again an algebraic conjugate of
an isotomic conjugate, and

hence of order 2.

The Cevapoint of P and U can be written P((U/P)^{-})^{-1} or
U((P/U)^{-})^{-1}.

Either way it is a congugate of the map sending Q to (Q^{-})^{-1}.

The crosspoint of P and U can be written P((U/P)^{-1})^{-} or
U((P/U)^{-1})^{-}.

Either way it is a congugate of the map sending Q to (Q^{-1})^{-}.

**15. The Steiner ellipses.**

We know that the Steiner Inellipse **S'** is the complement of the Steiner Ellipse **S**.

Suppose that U =[u,v,w] lies on **S**. Then U^{-} = [v+w,w+u,u+v] lies on **S'**.

Let G = [1,1,1]. Normalizing as necessary, UG/GU^{-} = 2, Thus U^{-} is the point

on **S'** "diametrically opposite" U. The other intersections with UG are given by

[2v+2w-u,2u+2w-v,2u+2v-w] on **S** and [4u+v+w,u+4v+w,u+v+4w] on **S'**.

As above [P,P^{-},G,P^{+}] = -1, so PG/GP^{+} = 1/2.

The point P lies inside **S'** if and only if its anticomplement lies inside **S**.

**16. Locating the centres of circumellipses.**

In **4**, we gave an algebraic characterization of the centre of a circumellipse.

We also gave a geometrical description of the perspector as an interior point

of the Steiner Inellipse **S'**. Here we find the regions in which the centre lies.

We know that U = [u,v,w] is the centre of a circumellipse if and only if

the perspector U* = [u*,v*,w*] lies within **S'**, i.e. F(u*,v*,w*) < 0, where

F is given in **4**. We have u* = u(v+w-u), v* = v(w+u-v), w* = w(u+v-w),

so that F(u,v,w) = -(u*+v*+w*).

Now we see that the condition that U be the centre of a circumellipse can be

expressed as u**+v**+w** > 0.

Multiplying out, u**+v**+w** = (u+v+w)(u+v-w)(u+w-v)(v+w-u).

Note that F(U*) = F(U^{2}).

As U is the centre of a circumellipse, U* lies in **S'** so can be taken

with positive coordinates. Then U* = V^{2}. There are essentially four

choices of V. Then F(V*) = f(V^{2}) = F(U*). Thus V is the centre of a

further circumellipse. Even if U is a centre, it is not clear that we can

choose V as a triangle centre. This relates to **4**.

If we normalise U so that u+v+w > 0, then the condition can be expressed

as either (u+v-w), (u+w-v), (v+w-u) are all positive, or exactly two are

negative.

Consider the lines x+y-z = 0, x-y+z = 0, -x+y+z = 0. These are the lines

through the mid-points of two of the sides of ΔXYZ so parallel to the sidelines.

They divide the plane into seven regions.

If U is in the finite (triangular) region, then all of our factors are positive.

If U is in a region which contains a vertex, then exactly one factor is negative.

For example, in the region containing X, only (v+w-u) is negative.

Thus U will be the centre of a circumellipse if and only if it lies in the

interior of a region which does not contain a vertex of the triangle.