[1] The projections of a point P of the circumcircle of triangle ABC on its sides are aligned on a line W(P) (see Simson.html ).

[2] The

[3] As P varies on the circumcircle lines W(P) envelope an algebraic curve of degree four called

[4] The orthopole P

In the most general case hypocycloids are curves generated by rolling a circle of radius b inside a circle of radius a > b.

Taking the center E of the fixed circle as origin, the x-axis in the direction of EA

The angle(P''QP

In the case of the

This equilateral is defined as follows: (i) Let A

It is an easy exercise to show that triangle A

[1] These parallels {EA

[2] The deltoid is tangent to the Euler circle.

[3] The cusps of the deltoid are on a circle of radius three times that of the Euler circle.

[4] The triangle formed by the cusps of the deltoid and the equilateral derivative are two homothetic triangles. The homothety-center is a point H' lying on the Euler line OE of ABC and being symmetric to the orthocenter H of ABC with respect to the circumcenter O (H' is X(20) or the De Longchamps point of triangle ABC). The homothety-ratio is the homothety ratio of the circumscribing circles i.e. 3/2.

The figure is best analysed by starting with point P on the circumcircle of ABC. Its Wallace line W(P) intersects the Euler circle at a point P' which is the middle of the segment PH joining P to the orthocenter of ABC. This basic fact (see SteinerLine.html ) implies that taking the symmetric of the Euler circle with respect to P' we produce a second circle (the rolling one) with radius equal to the radius of the Euler circle and tangent to it at P'.

[1] The quadrilateral EHQP is a parallelogram since P' is common middle of its diagonals PH and EQ.

[2] Segment QP is always parallel to the Euler line OE and has fixed length equal to OH.

[3] When the Wallace line is W(P) = EA

[4] Since two Wallace lines are inclined to each other by half the corresponding central angle of their defining points (see SimsonProperty2.html ) we have angle(A

Here w = angle(W(P),A

When W(P) obtains the position orthogonal to t

[1] The symmetric P

[2] P

[3] P

[4] P

[5] The orthopoles P

[6] P

[7] Thus P on the circumcircle determines a unique right-angled triangle P

[8] The right-angled triangles PP

[9] Since, by appropriate selection of P, P

[10] Point P

[11] Thus P

[12] The middles of segments P

[13] The third Wallace line through P

[14] PH

[2] P coincides with C. Then W(P) coincides with the altitude to AB, points {P

[2] Gallatly William

[3] Honsberger, R.

[4] Goormaghtigh, R.

[5] Karl Cordia Mary

[6] Ramler, J. O.

[7] Van Horn, C. E.

DeltoidBasic.html

Euler.html

Hypocycloid.html

Orthopole.html

Orthopole2.html

S_Triangles.html

Simson.html

SimsonProperty2.html

SteinerLine.html

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