MathDB

3

Part of 2009 Bundeswettbewerb Mathematik

Problems(2)

concurrency related to symmetric wrt side midpoints

Source: Germany Federal - Bundeswettbewerb Mathematik 2009, round 1, p3

4/9/2020
Let PP be a point inside the triangle ABCABC and Pa,Pb,PcP_a, P_b ,P_c be the symmetric points wrt the midpoints of the sides BC,CA,ABBC, CA,AB respectively. Prove that that the lines APa,BPbAP_a, BP_b and CPcCP_c are concurrent.
geometryconcurrencyconcurrentmidpointsSymmetric
PQ/CQ<= [AB/(AC+CB)]^2 , angle bisector and circumcircle related

Source: Germany Federal - Bundeswettbewerb Mathematik 2009, round 2, p3

4/9/2020
Given a triangle ABCABC and a point PP on the side ABAB . Let QQ be the intersection of the straight line CPCP (different from CC) with the circumcicle of the triangle. Prove the inequality PQ‾CQ‾≤(AB‾AC‾+CB‾)2\frac{\overline{PQ}}{\overline{CQ}} \le \left(\frac{\overline{AB}}{\overline{AC}+\overline{CB}}\right)^2 and that equality holds if and only if the CPCP is bisector of the angle ACBACB. https://cdn.artofproblemsolving.com/attachments/b/1/068fafd5564e77930160115a1cd409c4fdbf61.png
geometryangle bisectorcircumcirclegeometric inequality