MathDB

3

Part of 1997 Bundeswettbewerb Mathematik

Problems(2)

ST \ge 2r(\sqrt{2}-1) , inscribed circles in a semicircle

Source: 1997 German Federal - Bundeswettbewerb Mathematik - BWM - Round 2 p3

1/27/2020
A semicircle with diameter AB=2rAB = 2r is divided into two sectors by an arbitrary radius. To each of the sectors a circle is inscribed. These two circles touch ABB at SS and TT. Show that ST≥2r(2−1)ST \ge 2r(\sqrt{2}-1).
geometrycirclessemicircle
congruent squares inscribed in a triangle

Source: 1997 German Federal - Bundeswettbewerb Mathematik - BWM - Round 1 p3

1/27/2020
A square SaS_a is inscribed in an acute-angled triangle ABCABC with two vertices on side BCBC and one on each of sides ACAC and ABAB. Squares SbS_b and ScS_c are analogously inscribed in the triangle. For which triangles are the squares Sa,SbS_a,S_b, and ScS_c congruent?
geometrySquarescongruent