Home page | Catalogue | Classes | Tables | Glossary | Notations | Links | Bibliography | Thanks | Downloads | Related Curves

(a^4 + b^2c^2) x (c^2 y^2 - b^2 z^2) = 0

(b^2 x - c^2 y) + (b^2 z - c^2 x) = 0

X(1), X(3), X(4), X(32), X(39), X(76), X(83), X(194), X(384), X(695), X(2896), X(3491) up to X(3503), X(14370)

excenters, vertices of cevian triangle of X(384)

vertices A1, B1, C1 and A3, B3, C3 of the first and third Brocard triangles

reflections A1', B1', C1' of A1, B1, C1 in BC, CA, AB and their isogonal conjugates. These points lie on the cevians of X(83), X(39) respectively. A1, B1, C1 and A1', B1', C1' are the vertices of the Kiepert triangles with base angle Ω and -Ω respectively. See Table 32 and also K797.

K020 is the isogonal pK with pivot X(384) hence a member of the Euler pencil. See Table 27.

Locus properties :

Let A1B1C1 be the first Brocard triangle and M a variable point. The lines MA1, MB1, MC1 meet BC, CA, AB at Ma, Mb, Mc respectively. These three points are collinear if and only if M lies on K017. They form a triangle perspective to ABC if and only if M lies on K020. This is also true with the third Brocard triangle. In the case of the first Brocard triangle, the locus of the perspector is K322 and for the third Brocard triangle, the locus of the perspector is K532. Further details and figure below.

See also a generalization at CL041.

The isotomic transform of K020 is K743.

 

K020Brocard1 K020Brocard3

K020 is a pK with pivot X(32) and isopivot X(3493) with respect to the first Brocard triangle.

X(3493) is the tangential of X(32) in K020 and also the perspector of the first Brocard triangle and the anticevian triangle of X(694).

K020 is a pK with pivot X(76) with respect to the third Brocard triangle. Its isopivot is X(8871) = X(76)', the tangential of X(76) in K020.

X(8871) is also the perspector of the third Brocard triangle and the anticevian triangle of X(694).

 

K020BR1BR3

Let Q be a variable point on K020. Denote by BR1 = A1B1C1 and BR3 = A3B3C3 the 1st and 3rd Brocard triangles.

• If P1a = QA1 /\ BC, P1b = QB1 /\ CA, P1c = QC1 /\ AB then P1aP1bP1c is the cevian triangle (with respect to ABC) of a point P1which lies on K322 and P1aP1bP1c, BR1 are perspective at Q. Recall that ABC and BR1 are perspective et X(76).

• Similarly with BR3 instead of BR1, we find P3 on K532 such that the cevian triangle P3aP3bP3c of P3 is perspective at Q to BR3. Recall that ABC and BR3 are perspective at X(32).

• Let P4 be the P3-Ceva conjugate of Q and P2 the X(385)-Ceva conjugate of P4. These points lie on K128 and they are collinear with X(694), the isopivot of K128, hence they are X(385)-Ceva conjugate points since X(385) is the pivot of K128.

The anticevian triangle of P2 (resp. P4) is perspective at Q to BR1 (resp. BR3).

Note the following collinearities :

Q, P1, P3, X(384) – Q, P2, X(32) – Q, P4, X(76) and X(385), P1, P2 – X(385), P3, P4. Also X(694), P2, P4 as already said.

The following table gives the correspondences between Q and the four points P1, P2, P3, P4.

Q on K020

P1 on K322

P3 on K532

P2 on K128

P4 on K128

1

335

1911

3509

3510

3

2

237

6

3511

4

297

25

 

3186

32

385

32

3506

32

39

1916

9468

511

3229

76

76

385

76

 

83

 

733

 

 

194

698

6

 

2

384

694

694

385

385

695

 

 

3505

 

2896

141

2076

2

 

3224

2998

 

3504

 

3491

 

 

3229

511

3492

384

 

32

3506

3493

 

 

694

 

3494

 

 

2319

 

3495

 

 

 

 

3496

257

 

1

 

3497

 

 

 

 

3498

 

 

 

 

3499

 

3051

3511

6

3500

 

 

 

 

3501

 

 

3507

3508

3502

 

 

 

 

3503

 

 

 

1423

6196

 

904

 

1

7346

 

 

 

 

8790

 

384

 

76

8861

 

 

98

 

8862

 

 

1423

 

8863

 

 

3186

 

8864

 

 

3225

 

8865

 

 

3508

3507

8866

 

 

3510

3509

8867

 

 

3512

 

8868

 

 

7168

 

8870

 

 

 

98

8871

 

 

 

694

8872

 

 

 

2319

8873

 

 

 

3225

8874

 

 

 

3505

8875

 

 

 

3512

8876

 

 

 

7166

384/3224

 

 

 

3504

384/8868

 

 

 

7168

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Notes

1. when Q is the pivot X(384) of K020, the points P1, P2, P3, P4 are the pivots of their corresponding cubic (pink cells). These pivots X(385), X(694) lie on the three cubics.

2. two points Q on K020 collinear with X(695) – the isopivot of K020 – hence X(384)-Ceva conjugates correspond to swapped X(385)-Ceva conjugate points P2, P4 on K128. See for example the yellow and orange lines.

3. two isogonal conjugate points Q, Q* on K020 hence collinear with X(384) correspond to :

• the pairs {P1, P1'} on K322 and {P3, P3'} on K532 also collinear with X(384).

• P2 and P2' on K128 on a line passing through the isogonal conjugate of X(3505), a point obviously on K128.

• P4 and P4' on K128 on a line passing through a fixed point on K128 with first barycentric coordinate :

(a^4+b^2 c^2)/(a^6 b^6-a^4 b^4 c^4+a^6 c^6-b^6 c^6)

and SEARCH = -3.67358153099718.

This point is the isogonal conjugate of the X(385)-Ceva conjugate of X(3505).