Exercise 37.11
Verify the following ruler-only construction of the inverse of a point A (7 steps): Draw OA, get R, S. Draw any line I through A meeting Г in P,Q.
Draw RP and SQ to meet at T. Draw RQ and PS to meet at U. Draw TU to meet OA at A'. Show also that TU is perpendicular to OA.
Consider the triangle RST; the lines PS and RQ are altitudes; therefore also TU is an altitude, hence it is orthogonal to the line OA.
In the construction the angles QRS and QPS are congruent by III.21
Since is a right angle we get that PUR =
The points R,P,A',U lie on a circle since the angles RPU and RA'U are congruent. It follows that RUP and RA'P are congruent, and the triangles OPA, OA'P are similar. From this we get OA x OA' = OP.