Dear All My Friends,

Thank you very much for your references and remarks.

I have some results:

- L bound with ABC one complete quadrilateral with Miquel point = X(110)

- Three lines connected A, B, C with reps Miquel circle centers are concurrent at X(477)

- Midpoint of QR = X(1511)

Best regards,

Bui Quang Tuan

Paul Yiu <

yiu@...> wrote: Dear friends,

BQT (14935) Given triangle ABC with circumcircle (O). There is and

exactly one equilateral triangle inscribed in (O) taken A as one

vertex. We denote its bottom line wrt A as L_a, intesection of L_a

with sideline BC as A'. Similarly define L_b, B', L_c, C'. Three lines

L_a, L_b, L_c bound one triangle A''B''C''.

(1) ABC and A''B''C'' are perspective at a point P on the circumcircle

(O).

(2) A', B', C' are collinear on one line L.

(3) If L intersects (O) at two points Q, R, then PQR is equilateral.

PY (14936) How wonderful. I make a quick sketch and see that your

P is the isogonal conjugate of the infinite point of the Euler

line.

*** The line L_a is the perpendicular to OA at the point which

divides OA in the ratio 3:-1.

FR (14939) I think the point P is known for a long time. It is X(74).

It was found by Dobbs more than 100 years ago with just your way!

APH (14940) We inscribe in (O) THE isosceles triangle AB^*C^* with

A = \omega , B* = C* = \frac {\pi-\omega}2. Denote the line B^*C^*

as L_a. Similarly the lines L_b, L_c.

Is the Triangle bounded by the lines L_a, L_b, L_c perspective

with ABC for every given angle \omega (with 0 < \omega < \pi) ?

*** Yes, the locus is the Jerabek hyperbola. If L_a is the

perpendicular to OA at the point which divides AO in the ratio t:1-t,

this perspector is the point

(\frac {a^2}{S^2-t a^2S_A}: ... : ...).

PN (14944, 14946):

Let ABC be a triangle with circumcircle (O) and the incircle (I).

C_a is the circle with center A and radius AI. L_a is the radical

axis of the circles C_a and (O). A' is the intersection point of

the lines L_a and BC. Similarly define C_b, L_b, B', C_c, L_c, C'.

Three lines L_a, L_b, L_c bound one triangle A"B"C".

(1) L_a is tangent to the incircle (I).

(2) A', B', C' are collinear points.

(3) AA", BB" and CC" are concurrent lines.

(4) The point P lies on the line OI.

*** This is X(65), the intersection of OI with the Jerabek hyperbola.

It is also the isogonal conjugate of the Schiffler point,

(5) OI and L are perpendicular lines.

*** This is the polar wrt to (O) of the exsimilicenter of (O) and (I).

Best regards

Sincerely

Paul

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