MathDB

3

Part of 2022 Bundeswettbewerb Mathematik

Problems(2)

A rather famous lemma in geometry

Source: Bundeswettbewerb Mathematik 2022, Round 1 - Problem 3

3/30/2022
A circle kk touches a larger circle KK from inside in a point PP. Let QQ be point on kk different from PP. The line tangent to kk at QQ intersects KK in AA and BB. Show that the line PQPQ bisects ∠APB\angle APB.
geometrytangentangle bisectorgeometry solvedtangent circlescurvilinear incircles
An isogonal conjugate problem

Source: German Bundeswettbewerb 2022, Round 2, Problem 3

9/14/2022
In an acute triangle ABCABC with AC<BCAC<BC, lines mam_a and mbm_b are the perpendicular bisectors of sides BCBC and ACAC, respectively. Further, let McM_c be the midpoint of side ABAB. The Median CMcCM_c intersects mam_a in point SaS_a and mbm_b in point SbS_b; the lines ASbAS_b und BSaBS_a intersect in point KK. Prove: ∠ACMc=∠KCB\angle ACM_c = \angle KCB.
geometryperpendicular bisector