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lines connecting symmetric of feet of altitudes wrt other altitudes are parallel

Source: 2015 Sharygin Geometry Olympiad Correspondence Round P6

August 2, 2018

geometryaltitudessymmetry

Problem Statement

Let

AA', BB'

and

CC'

be the altitudes of an acute-angled triangle

ABC

. Points

C_a, C_b

are symmetric to

C'

wrt

AA'

and

BB'

. Points

A_b, A_c, B_c, B_a

are defined similarly. Prove that lines

A_bB_a, B_cC_b

and

C_aA_c

are parallel.