MathDB

g2

Part of Indonesia MO Shortlist - geometry

Problems(3)

RS = \sqrt2 SQ if AD=CR , CR _|_ AB, ABC right isosceles

Source: Indonesia INAMO Shortlist 2008 G2

8/25/2021
Let ABCABC be an isosceles triangle right at CC and PP any point on CBCB. Let also QQ be the midpoint of ABAB and R,SR, S be the points on APAP such that CRCR is perpendicular to APAP and ∣AS∣=∣CR∣|AS|=|CR|. Prove that the ∣RS∣=2∣SQ∣|RS| = \sqrt2 |SQ|.
isoscelesequal segmentsgeometry
<BZC =90^o wanted, touchpoints with incircle, _|_ bisector

Source: Indonesia INAMO Shortlist 2010 G2

8/27/2021
Given an acute triangle ABCABC. The inscribed circle of triangle ABCABC is tangent to ABAB and ACAC at XX and YY respectively. Let CHCH be the altitude. The perpendicular bisector of the segment CHCH intersects the line XYXY at ZZ. Prove that ∠BZC=90o.\angle BZC = 90^o.
geometryincircleright angle
intersection points of common tangents of 2 circles are concyclic

Source: Indonesia INAMO Shortlist 2017 G2 https://artofproblemsolving.com/community/c1101409_indonesia_shortlist__geometry

11/15/2021
It is known that two circles have centers at PP and QQ. Prove that the intersection points of the two internal common tangents of the two circles with their two external common tangents lie on the same circle.
Concycliccommon tangentscommon tangentgeometry