MathDB

g3

Part of Indonesia MO Shortlist - geometry

Problems(4)

R^2=OE^2+CD^2 [1- BC^2/(AB^2+AC^2) ], circle tangent to (ABC)

Source: Indonesia INAMO Shortlist 2008 G3

8/25/2021
Given triangle ABCABC. A circle Γ\Gamma is tangent to the circumcircle of triangle ABCABC at AA and tangent to BCBC at DD. Let EE be the intersection of circle Γ\Gamma and ACAC. Prove that R2=OE2+CD2(1−BC2AB2+AC2)R^2=OE^2+CD^2\left(1- \frac{BC^2}{AB^2+AC^2}\right) where OO is the center of the circumcircle of triangle ABCABC, with radius RR.
geometrytangent circles
BE = BF wanted, cyclic ABCD

Source: Indonesia INAMO Shortlist 2009 G3 https://artofproblemsolving.com/community/c1101409_

12/11/2021
Given a quadrilateral ABCDABCD inscribed in circle Γ\Gamma.From a point P outside Γ\Gamma, draw tangents PAPA and PBPB with AA and BB as touspoints. The line PCPC intersects Γ\Gamma at point DD. Draw a line through BB parallel to PAPA, this line intersects ACAC and ADAD at points EE and FF respectively. Prove that BE=BFBE = BF.
geometryequal segmentscyclic quadrilateral
congruent triangles wanted, starting with intersecting circles

Source: Indonesia INAMO Shortlist 2010 G3

8/27/2021
Suppose L1L_1 is a circle with center OO, and L2L_2 is a circle with center O′O'. The circles intersect at A A and B B such that ∠OAO′=90o\angle OAO' = 90^o. Suppose that point XX lies on the circumcircle of triangle OABOAB, but lies inside L2L_2. Let the extension of OXOX intersect L1L_1 at YY and ZZ. Let the extension of O′XO'X intersect L2L_2 at WW and VV . Prove that △XWZ\vartriangle XWZ is congruent with △XYV\vartriangle XYV.
geometrycongruent trianglescircles
PQ//BC wanted, projections of A on angle bisectors

Source: Indonesia INAMO Shortlist 2017 G3 https://artofproblemsolving.com/community/c1101409_indonesia_shortlist__geometry

11/15/2021
In triangle ABCABC, points PP and QQ are projections of point AA onto the bisectors of angles ABCABC and ACBACB, respectively. Prove that PQ∥BCPQ\parallel BC.
geometryparallel