MathDB

2

Part of 1985 Bundeswettbewerb Mathematik

Problems(2)

4 projections of foot of altitude are collinear

Source: 1985 German Federal - Bundeswettbewerb Mathematik - BWM - Round 1 p2

11/21/2022
Prove that in every triangle for each of its altitudes: If you project the foof of one altitude on the other two altitudes and on the other two sides of the triangle, those four projections lie on the same line.
geometrycollinearprojectionsaltitude
r1 + r2 + r3 + r4 = 2r for radii of inspheres of terrahedron

Source: 1985 German Federal - Bundeswettbewerb Mathematik - BWM - Round 2 p2

11/21/2022
The insphere of any tetrahedron has radius rr. The four tangential planes parallel to the side faces of the tetrahedron cut from the tetrahedron four smaller tetrahedrons whose in-sphere radii are r1,r2,r3r_1, r_2, r_3 and r4r_4. Prove that r1+r2+r3+r4=2rr_1 + r_2 + r_3 + r_4 = 2r
geometry3D geometrytetrahedroninradiusSpheres