Construct first the orthopole S of the chord TT'. For this draw AR orthogonal to TT', R being its second intersection point with the circumcircle. Let R' be the projection of R on side BC. The orthopoles of all lines L parallel to TT' result by translating parallel the segment RR' so that R coincides with a point of line L (see Orthopole.html ). Thus the orthopole of TT' is point S, such that PS is parallel and equal to RR', P being the intersection point of TT' with AR.

Construct now the Wallace-Simson line W(T) of T. For this project T on BC to point X and extend to define its second intersection point T

This follows easily by an angle chasing argument using the cyclic quadrilateral TRPR

[1] Consider the intersection P

[2] Points O

[3] Obviously the reverse property holds also true. If three Wallace lines {W(Q

[4] Last assertion is symmetric with respect to the three points {Q

[5] It follows that S is the intersection point of two orthogonal lines, the first from Q

[6] Hence the three Wallace lines {W(Q

[7] It follows that O

[8] Letting {Q

[9] Because {P,P'} are diametral points the two Wallace lines {W(P),W(P')} intersect on the Euler circle (see SimsonDiametral.html ) and define on it a triangle R

[10] It follows further that {R

[11] From the aforementioned homothety follows that S is symmetric to the orthocenter H of ABC with respect to the middle Q of the segment O

[12] Lines W(S) envelope a deltoid studied in Deltoid.html .

[1] The orthopoles O

[2] The orthopole O

[3] The radius OP is parallel to the radius EP

[4] Points {M

These and the properties of the previous section are used in the discussion of the enveloping deltoid of the Wallace lines of the triangle ( Deltoid.html ).

Consider the Wallace lines W(P) and W(Q) of two points P, Q on the circumcircle lying closely to each other. Their intersection point O(PQ) was identified as the orthopole of line PQ. As Q tends to coincide with P, O(PQ) tends to O

[2] Gallatly William

[3] Goormaghtigh, R.

[4] Honsberger, R.

[5] Lalesco, T.

[6] Ramler, J. O.

[7] Van Horn, C. E.

Deltoid.html

Euler.html

Orthopole.html

S_Triangles.html

SimsonDiametral.html

SimsonProperty.html

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