MathDB

4

Part of 2010 Oral Moscow Geometry Olympiad

Problems(2)

a right-angled triangle can be constructed from the segments AK, CM,MK

Source: 2010 Oral Moscow Geometry Olympiad grades 8-9 p4

8/16/2020
An isosceles triangle ABCABC with base ACAC is given. Point HH is the intersection of altitudes. On the sides ABAB and BCBC, points MM and KK are selected, respectively, so that the angle KMHKMH is right. Prove that a right-angled triangle can be constructed from the segments AK,CMAK, CM and MKMK.
geometrytriangle inequalityright triangleisosceles
perpendicular wanted, perpendiculars inside parallelogram

Source: 2010 Oral Moscow Geometry Olympiad grades 10-11 p4

5/8/2020
From the vertex AA of the parallelogram ABCDABCD, the perpendiculars AM,ANAM,AN on sides BC,CDBC,CD respectively. PP is the intersection point of BNBN and DMDM. Prove that the lines APAP and MNMN are perpendicular.
geometryparallelogramperpendicular