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4 lines concurrent, in an interior point, starting with insphere of tetrahedron

Source: 2015 Sharygin Geometry Olympiad Correspondence Round 24

August 2, 2018

spheretetrahedron3-Dimensional Geometrygeometry3D geometry

Problem Statement

The insphere of a tetrahedron ABCD with center

O

touches its faces at points

A_1,B_1,C_1

and

D_1

. a) Let

P_a

be a point such that its reflections in lines

OB,OC

and

OD

lie on plane

BCD

. Points

P_b, P_c

and

P_d

are defined similarly. Prove that lines

A_1P_a,B_1P_b,C_1P_c

and

D_1P_d

concur at some point

P

. b) Let

I

be the incenter of

A_1B_1C_1D_1

and

A_2

the common point of line

A_1I

with plane

B_1C_1D_1

. Points

B_2, C_2, D_2

are defined similarly. Prove that

P

lies inside

A_2B_2C_2D_2

.

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