Miquel point and isogonal conjugation

X Y Z -that set is called family ( ) , , X Y Z of Miquel triangles.

We denote the interior angles of triangle XYZ △ with

, respectively.

1, .84 p For the Miquel point are in force these equations with directed angles:

Figure 1b Figure 1c A

All Miquel triangles of any fixed point P relative to triangle ABC △ are similar to themselves. One of these triangles is the pedal triangle of point P .

Given the basic triangle ABC . In this section we study the positions of point P , which Miquel triangle XYZ is similar to the given one. We make a thorough study of non-trivial cases -for scalene triangle and non-right triangle ABC , and point P is not on any of the line AB , BC and CA .

In the process of studying the positions of Miquel point, which Miquel triangles are similar to the given one, we use the following proposition. This means that finding all Miquel points, which triands are triangles similar to the given, will be comprehensively if all Miquel points in the circum-circle of triangle ABC are found.

Therefore we will consider the positions of the Miquel point in the circum-circle. We will use two important lemmas.

(i)

. Consequently in this case point P is also exterior for the Miquel triangle XYZ △ .

For the sake of brevity and convenience we will give some symbols:

Given triangle ABC and point P . Then denote with directed angles (Figure 2a,2b):

Given point P interior for the circum-circle of triangle ABC and XYZ is Miquel triangle of point P relative to triangle ABC △ . Then the following equations for the angles of triangle XYZ △ are in force:

Proof. Equations with directed angles we prove using inscribed angles with the same adjacent arc. In Figure 2a is the case where the point P is in the interior of triangle ABC △

. In Figure 2b is the case where point P is in the exterior of triangle ABC △ , but in the circum-circle of triangle ABC -now 2 β and 1 γ have negative digits.

∢ . We consider the similarities separately applying Lemma 2.

These equations determine that in this case point P coincides with point O -the center of the circum-circle of triangle ABC △ .

α β γ = = and now point P coincides with the First Brocard point of triangle β γ γ γ

-here point P coincides with the Second Brocard point of triangle ABC △ .

In the next three cases we use some facts for the symmedians in a triangle.

Definition 3. Symmedian of a triangle is a line that is symmetrical to the median with respect to the angle bisector of the interior angle of the triangle. . In Figure 3 it is shown the case 90 A < °∢ , and in Figure 4 -90 A > °∢ . First we consider the case 90

Consequently, if exists, point P is on the arc from which the side BC is visible at angle 2 A ∢ , then point P is also on the arc from which the side CA is visible ate angle 180 A °-∢ , and point P is on the arc from which the side AB is visible at angle Taking account of that for each point P its pedal triangle is one of the family of triangles XYZ △ , and the pedal triangles of the inverse points with respect to the circum-circle are similar, we some to the conclusion that in the plane of a given triangle there are 5 points, which Miquel triangles are similar to the given one -these points are the inverse points with respect to the circum-circle of the first Brocard point, the second Brocard point and the points A S , B S and C S , since the inverse point of the center of the circum-circle is an infinite point.

We will consider concrete positions of the Miquel point starting with the positions for which Miquel triangles are similar to the main triangle. Proof. We will consider two cases:

(1) Let point P is orthocenter of the acute-angled triangle ABC △ (Figure 6a).

∢ . We will use the received symbols of the angles of the triangles and we will consider two cases -if 90 A < °∢ and if 90 A > °∢ .

Figure 7b (1) If 90 A < °∢ (Figure 8). Then ( )

From the equations of inscribed angles It follows that

. That means that the quadrangle PYFZ is a parallelogram and XE is a median of triangle XYZ △ .

(2) If 90 A > °∢ (Figure 9). We extend XP till it intersects the Miquel circle through vertex A in point F . We have 1 Proof. We also consider two cases.

(1) If triangle ABC △ is acute-angled (Figure 10). (2) If the triangle is obtuse-angled one-let 90 A > °∢ (Figure 11). In this case point P is exterior for triangle ABC △ , it is on the arc from the circle through points (2) If 90 A > °∢ . In this case point P coincides with X S in XYZ △ , which is shown in Figure 12b.

Between the considered positions of Miquel point there i