MathDB

1

Part of 2016 Czech-Polish-Slovak Junior Match

Problems(2)

largest possible area of the quadrilateral MKCL

Source: Czech-Polish-Slovak Junior Match 2016, individual p1 CPSJ

1/20/2020
Let ABAB be a given segment and MM be its midpoint. We consider the set of right-angled triangles ABCABC with hypotenuses ABAB. Denote by DD the foot of the altitude from CC. Let KK and LL be feet of perpendiculars from DD to the legs BCBC and ACAC, respectively. Determine the largest possible area of the quadrilateral MKCLMKCL.
Czech Republic
geometryareamaxperpendicular
angle chasing candidate, <AP B + < QCR = 180^o , 3 midpoints related

Source: Czech-Polish-Slovak Junior Match 2016, team p1 CPSJ

1/20/2020
Let ABCABC be a right-angled triangle with hypotenuse ABAB. Denote by DD the foot of the altitude from CC. Let Q,RQ, R, and PP be the midpoints of the segments AD,BDAD, BD, and CDCD, respectively. Prove that ∠APB+∠QCR=180o\angle AP B + \angle QCR = 180^o.
Czech Republic
geometrymidpointsanglesAngle Chasing