Theorem (William Wallace, 1797)

The pedal triangle of a point P with respect to ABC degenerates into a straight line iff P lies on the circumcircle of ABC.

(The line is known as Simson line, or just plain simson. The theorem is often called Simson-Wallace Theorem.)

With a reference to the above diagram (other cases of relative point positions are similar), let P lie on the circumcircle of ABC. Because of the right angles at A', B' and C' there are several cyclic quadrilaterals. In other words, P lies on the circumcircles of triangles A'BC', A'B'C and AB'C'. Since P is concyclic with (of course) ABC and A'BC',

APC = B = A'PC'.

By adding CPC', we obtain

A'PC = APC'.

P is also concyclic with A', C and B'. Therefore,

A'PC = A'B'C.

Similarly, since P is concyclic with A, B' and C',

APC' = AB'C'.

Thus

A'B'C = AB'C',

which exactly means that the points A', B' and C' are collinear - the pedal triangle degenerates into a line. And it should be noted that the steps in the above proof are reversible, provided the points are named in an appropriate way.