Let ABC be a triangle.
Let the point P lie in the plane of ABC.
Let X, Y, Z be the feet of the perpendiculars from P to the lines BC, AC, AB respectively.
PROVE: : The points X, Y , Z are collinear if and only if P lies on the circumcircle of ABC.
(In reply to re: Solution
Since the proposition states that Z is the foot of a perpendicular from P
to AB etc., it’s a moot point as to whether we include the special case
of P on AB. But clearly, if we do, and if P coincides with A, then so must
Y and Z, rendering the collinearity of XYZ trivial.
In the ‘converse’, the possibility of XYZ coinciding with a side, say BC,
is more interesting: we would then have Z & B coincident and Y & C
coincident so that ABPC (i.e. AZPY) is clearly cyclic.
You’re right Bractals, I should have included this second point in my
Posted by Harry
on 2013-12-24 13:18:38