Miquel_Point

1. Miquel pivot point

Select three points {A',B',C'} on the sides of triangle ABC as shown. Then circles (AB'C'), (BC'A'), (CA'B') pass all through a common point P.

For all possible relative positions of {A',B',C'} this follows by an easy angle-chasing argument as the one shown in the above figure.
Remark The property holds for all relative positions of {A',B',C'} on the respective lines {BC,CA,AB}. Even in the two cases in which :
(i) One point, C' say, coincides with one vertex, A say (first figure below).
(ii) The three points {A',B',C'} are on a line (second figure below).

The name pivot comes from the fact that another tripple of points {A'',B'',C''} correspondingly on {BC,CA,AB} defining, through the corresponding circles {(AB''C''),(BC''A''),(CA''B'')}, the same point P, defines a kind of "rotation" of the configuration.
This because the three triangles {PA'A'',PB'B'',PC'C''} are similar. Consequently the triangles A'B'C', A''B''C'' are also similar. Thus, for varying fi, one can consider that triangle A'B'C' is pivoting around P, its vertices gliding on the side-lines of ABC.

In the above figure C' is considered coiniciding with A. The proof of the property is again an easy angle-chasing argument.

Remark-1 The property has also another aspect: Consider two circles (ABA') intersecting BC at an arbitrary A' on BC and circle (CA'B') with B' arbitrary on AC, the two circles intersecting at P. Then the circle (APB') is tangent to AB at A.

Remark-2 And still another aspect: Consider a circle (APB') tangent to AB at A and B' arbitrary on AC, and circle (B'CA') with A' arbitrary on BC, the two circles intersecting at P. Then circle (APA') passes through B.

2. Miquel pivot point - the case of collinear points

Select three collinear points {A',B',C'} on the sides of triangle ABC as shown. Then circles (AB'C'), (BC'A'), (CA'B') pass all through a common point P.

Corollary-1 The circumcircle of ABC passes through P too.
In fact, in this case the figure has two aspects :
(i) triangle ABC + intersecting line A'B'C'.
(ii) triangle A'BC' + intersecting line AB'C.
In the first case we have coincidence at P of the circles (AB'C'), (BC'A') and (CA'B').
In the second case we have coincidence at P of the circles (A'B'C), (BCA) and (C'AB').

Corollary-2 Lines {PA',PB',PC'} are equally inclined to sides {BC,CA,AB} of the triangle.

A particular case of the last corollary is the one in which the three equal angles coincide with a right angle. Then line A'B'C' is a Simson-Wallace line of the triangle ABC. But a slight generalization follows also easily:

Corollary-3 (Generalization of the Simson-Wallace line) If P is on the circumcircle and {A',B',C'} on sides {BC,CA,AB} such that {PC',PA',PB'} are respectively equal inclined to these sides, then {A',B',C'} are collinear.
Inversely, if points {C',A',B'} are collinear and set respectively on the sides {AB,BC,CA} of triangle ABC, then there is a point P such that {PC',PA',PB'} respectively are equal inclined to these sides and P is on the circumcircle of ABC.

3. Miquel pivot point - the four lines viewpoint

The figure of the previous section can be viewed as a property of the quadrilateral:
The circumcircles of the four triangles formed by any three of the sides of a quadrilateral intersect at a point P. We call it the Miquel pivot point of the quadrangle.
To build such a triangle we leave out a side (line) of the quadrangle and build the triangle having sides the remaining lines. This figure has many properties and applications. Some of them are discussed below.

Property-1 Consider the four centers {O₁,O₂,O₃,O₄} of the four circumcircles respectively {c₁,c₂,c₃,c₄} resulting by the four triangles of a quadrangle, each defined by three lines of it.
Leaving one of the O_i out, the triangle formed by the other three is similar to the one circumscribed by c_i (i.e. the circle whose center was left out).

By the conceptual symmetry of the configuration it suffices to show it for one triangle, O₁O₂O₃ say. But this follows by remarking that PA' is orthogonal to O₁O₂ and analogous orthogonalities for the other sides, as shown.

Property-2 The four centers {O₁,O₂,O₃,O₄} of the four circumcircles lie on a circle c₀. This circle passes through P.

First assertion follows directly from property-1. For example, leaving out O₂, triangle O₁O₃O₄ is similar to triangle B'CA' and the angles at O₄ and O₂ sum up to two right angles.
The other assertion follows also easily by remarking, for example, that O₃PO₂ is similar to C'PA', hence the angles at P and O₁ sum up to two right angles.

Property-3 If quadrangle BCB'C' is cyclic then its circumcenter lies also on circle c₀.
This follows from an easy angle chasing argument and property-1.

4. The Simson-Wallace line of P

[1] The Simson-Wallace lines of the four triangles {ABC,AC'B',B'CA',A'C'B} created from quadrangle BCB'C' coincide with a line L.
[2] The orthocenters of these four triangles are collinear and their support-line L' is parallel to L and d(P,L')=2d(P,L).

First assertion follows immediately from the fact that two sides of the quadrangle participate both in two triangles out of the four and from corollary-3 of section 2.
Second assertion is a consequence of the first and the general fact that a line L' parallel to a Simson-Wallace line L(P) at double distance from P passes through the orthocenter of the respective triangle (see SteinerLine.html ).

5. The parabola tangent to four given lines

[1] There is exactly one parabola tangent to four lines in general position.
[2] The focus of the parabola is the Miquel point P of the quadrilateral formed by the sides of the quadrilateral. The tangent at the vertex of the parabola is the Simson-Wallace line of P with respect to any of the four triangles created by the sides of the quadrilateral.

[1] Is a consequence of the general property of conics to be uniquely defined through five lines in general position (i.e. no three of them passing through a point). The fifth missing line is the line at infinity to which every parabola is tangent.
[2] Is a consequence of the general property of triangles whose sides are tangents to a parabola. Their circumcircle passes through the focus and their Simson-Wallace line of the focus is the tangent at the vertex of the parabola (their orthocenter then lying on the directrix see ParabolaChords.html ).

Remark There is a short of "dual" Miquel-pivot, in which we consider three arbitrary lines passing through the vertices and a tripple of resulting circles intersecting again at a point. See the file MiquelDual.html for details.

1. Miquel pivot point

2. Miquel pivot point - the case of collinear points

3. Miquel pivot point - the four lines viewpoint

4. The Simson-Wallace line of P

5. The parabola tangent to four given lines

See Also