math 243 winter 2008 geometry ii transformation geometry solutions to final examination practice problems department of mathematical and statistical sciences university of alberta question 1 show that thomsen s relation ...
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...Math winter geometry ii transformation solutions to final examination practice problems department of mathematical and statistical sciences university alberta question show that thomsen s relation is still true if each reection x replaced by a halfturn b c are three distinct points in the plane then explain how proof must be changed solution note d fourth point such abcd parallelogram cd ab so thus product an involution therefore which given line following conditions equivalent e or f through perpendicular since suppose imply only xed on now let p arbitrary joining also however involutions has as implies for all inverse parallel midpoint ac bc segment using halfturns prove diagonals bisect other hint n diagonal bd where last equality follows from fact isometry any involutive we done not m have bisector maps onto itself...