ABOUT THIRD PEDAL QUADRANGLE

Abstract

It was proved in previous article that with respect of any point in the interior of any convex quadrangle the fourth pedal quadrangle is similar to the initial quadrangle. In the present work the following problem is discussed. Is there a point in the interior of any convex quadrangle such that the third pedal quadrangle with respect of this point is similar to the initial quadrangle? The answer is positive. It is established, that there is a point O in the interior of any convex quadrangle ABCD such that the third pedal quadrangle is similar to ABCD. In the second part of the present article, this result is applied for the class of semi canonical trapeziums. First, it is proved that the first, second, third pedal quadrangles with respect of midpoints of diagonals of semi canonical trapeziums are similar to the initial trapezium. Then it is established, that there is a point O on the middle perpendicular of the base of semi canonical trapezium ABCD such that the second pedal quadrangle with respect of this point is a square.