Solutions - Georg Mohr-Konkurrencen

|AX| = |AB| and |DY | = |DC|. Consequently,|AD| |AX| + |XY | + |Y D| = |AB| + |BC| √2+ |DC| .̸ ̸̸ ̸̸ ̸ ̸̸ ̸ ̸ADreplacementsYAXBCB N CDFigure 3E9. QAnswer: the locus of the points P is the union of the diagonals AC andPBD .Let A Q be a point such that P QCD is a parallelogram (see Figure 4). ThenBABQP is also a parallelogram. From the equality AP D + BP C = 180 ◦Cit follows that BQC + BP C = 180 ◦ , so the points B , Q, C , P lieDon a common circle. Therefore, P BC = P QC = P DC , and sinceM|BC| = |CD|, we obtain that CP B = CP D or CP B +̸ CP D = 180 ◦ .NHence, the point P lies on the segment AC or on the segment BD .KXDCDCPQPQABABFigure 4Conversely, any point P lying on the diagonal AC satisfies the equation̸ BP C = ̸ DP C . Therefore, ̸ AP D + ̸ BP C = 180 ◦ . Analogously, weshow that the last equation holds if the point P lies on the diagonal BD .10. Answer: ̸ BAC = 60 ◦ , ̸ ABC = 105 ◦ and ̸ ACB = 15 ◦ .Suppose the line AD meets the circumcircle of triangle ABC at A and E(see Figure 5). Let M be the midpoint of BC and O the circumcentre oftriangle ABC . Since the arcs BE and EC are equal, then the points O ,M , E are collinear and OE is perpendicular to BC . From the equality7