The Simson line of a triangle
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Given any triangle ABC and any point in the plane P, let X be the foot of the altitude from P to AB, let Y be the foot of the altitude from P to AC and let Z be the foot of the altitude from P to BC.
Simson's theorem then says that X, Y and Z are collinear if and only if P lies on the circumcircle of ABC.
In the diagram above, the points A, B and C are free to move (click and drag the blue dots). The point P is free to move on the circumcircle of ABC (click and drag the purple dot).
By moving the point P around the circumcircle, we see another beautiful result of Simson: as P moves all the way around the circle, the line through X, Y and Z (coloured red in the diagram above) goes through a half-rotation.
In the next diagram, the point P is allowed to move freely and the line XY is drawn in red. In this case you can see that the points X, Y and Z will not be collinear (i.e. the point Z will not lie on the line XY) if the point P is not on the circumcircle. You can see this by clicking and dragging the point P in the diagram below.
