MathDB

3

Part of 2010 Bundeswettbewerb Mathematik

Problems(2)

P, Q, R, S lie either on a circle or on one straight line

Source: Germany Federal - Bundeswettbewerb Mathematik 2010, round 2, p3

4/14/2020
Given an acute-angled triangle ABCABC. Let CBCB be the altitude and EE a random point on the line CDCD. Finally, let P,Q,RP, Q, R and SS are the projections of DD on the straight lines AC,AE,BEAC, AE, BE and BCBC. Prove that the points P,Q,RP, Q, R and SS lie either on a circle or on one straight line.
Concycliccollinearprojectionsaltitudeperpendiculargeometry
similar triangles externally to a triangle lead to a similar of circumcenters

Source: Germany Federal - Bundeswettbewerb Mathematik 2010, round 1, p3

4/10/2020
On the sides of a triangle XYZXYZ to the outside construct similar triangles YDZ,EXZ,YXFYDZ, EXZ ,YXF with circumcenters K,L,MK, L ,M respectively. Here are ∠ZDY=∠ZXE=∠FXY\angle ZDY = \angle ZXE = \angle FXY and ∠YZD=∠EZX=∠YFX\angle YZD = \angle EZX = \angle YFX. Show that the triangle KLMKLM is similar to the triangles . https://cdn.artofproblemsolving.com/attachments/e/f/fe0d0d941015d32007b6c00b76b253e9b45ca5.png
geometrysimilar trianglesCircumcentercircumcircle