Simson's theorem

Draw a triangle and its circumscribed circle. Take a fourth point and draw the perpendiculars to the triangle's sides. Connect their landing points. If and only if the fourth point is on the circumscribed circle do these points lie on a straight line.

Please enable Java for an interactive construction (with Cinderella).

On the left you see the 'if': drag the blue point along the circle and note the black straight line. You can also modify the triangle.

On the right the 'only if': drag the blue point and find that in general the three constructed points form a yellow triangle, with opposite handedness on either side of the circumcircle and continuously reshapable.

Created with Cinderella by Leo Dorst 20010118