proof of Simson’s line
Given a with a point on its circumcircle (other than ),we will prove that the feet of the perpendiculars
drawn from P to the sides (or their prolongations) are collinear
.
Since is perpendicular to and is perpendicular to the point lies on the circumcircle of .
By similar arguments, also lies on the circumcircle of and .
This implies that , and are all cyclic quadrilaterals.
Since is a cyclic quadrilateral,
implies
Also is a cyclic quadrilateral, therefore
(opposite angles in a cyclic quarilateral are supplementary).
From these two, we get
Subracting , we have
Now, since is a cyclic quadrilateral,
also, since is a cyclic quadrilateral,
Combining these two results with the previous one, we have
This implies that the points are collinear.