TPTP Problem File: GEO384^1.p
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% File : GEO384^1 : TPTP v9.0.0. Released v7.0.0.
% Domain : Geometry (Coordinates and vectors in 3D space)
% Problem : International Mathematical Olympiad, 1961, Problem 6
% Version : [Mat16] axioms : Especial.
% English : Consider a plane epsilon and three non-collinear points A, B, C
% on the same side of epsilon; suppose the plane determined by
% these three points is not parallel to epsilon. In plane epsilon
% take three arbitrary points A', B', C'. Let L, M, N be the
% midpoints of segments AA', BB', CC'; let G be the centroid of
% triangle LMN. (We will not consider positions of the points A',
% B', C' such that the points L, M, N do not form a triangle.)
% What is the locus of point G as A' , B' , C' range independently
% over the plane epsilon.
% Refs : [Mat16] Matsuzaki (2016), Email to Geoff Sutcliffe
% : [MI+16] Matsuzaki et al. (2016), Race against the Teens - Benc
% Source : [Mat16]
% Names : IMO-1961-6.p [Mat16]
% Status : Theorem
% Rating : ? v7.0.0
% Syntax : Number of formulae : 3494 ( 726 unt;1208 typ; 0 def)
% Number of atoms : 6790 (2217 equ; 0 cnn)
% Maximal formula atoms : 22 ( 2 avg)
% Number of connectives : 39655 ( 106 ~; 233 |;1188 &;36002 @)
% (1095 <=>;1031 =>; 0 <=; 0 <~>)
% Maximal formula depth : 33 ( 8 avg)
% Number arithmetic : 4467 ( 374 atm;1203 fun; 954 num;1936 var)
% Number of types : 40 ( 36 usr; 3 ari)
% Number of type conns : 2408 (2408 >; 0 *; 0 +; 0 <<)
% Number of symbols : 1227 (1184 usr; 81 con; 0-9 aty)
% Number of variables : 8067 ( 407 ^;7085 !; 439 ?;8067 :)
% ( 136 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_ARI
% Comments : Theory: RCF; Score: 7; Author: Jumma Kudo;
% Generated: 2014-12-16
% : Answer
% ^ [V_Gs_dot_0: '3d.Shape'] :
% ( ~ ( '3d.colinear/3' @ ( '3d.point/3' @ 'Ax/0' @ 'Ay/0' @ 'Az/0' ) @ ( '3d.point/3' @ 'Bx/0' @ 'By/0' @ 'Bz/0' ) @ ( '3d.point/3' @ 'Cx/0' @ 'Cy/0' @ 'Cz/0' ) )
% & ( ( 'Az/0' != 'Bz/0' )
% | ( 'Bz/0' != 'Cz/0' )
% | ( 'Cz/0' != 'Az/0' ) )
% & ( $less @ 0.0 @ 'Az/0' )
% & ( $less @ 0.0 @ 'Bz/0' )
% & ( $less @ 0.0 @ 'Cz/0' )
% & ( V_Gs_dot_0
% = ( '3d.shape-of-cpfun/1'
% @ ^ [V_G_dot_0: '3d.Point'] :
% ( ( '3d.z-coord/1' @ V_G_dot_0 )
% = ( $quotient @ ( $sum @ 'Az/0' @ ( $sum @ 'Bz/0' @ 'Cz/0' ) ) @ 6.0 ) ) ) ) ) )
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include('Axioms/MAT001^0.ax').
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thf('Ax/0_type',type,
'Ax/0': $real ).
thf('Ay/0_type',type,
'Ay/0': $real ).
thf('Az/0_type',type,
'Az/0': $real ).
thf('Bx/0_type',type,
'Bx/0': $real ).
thf('By/0_type',type,
'By/0': $real ).
thf('Bz/0_type',type,
'Bz/0': $real ).
thf('Cx/0_type',type,
'Cx/0': $real ).
thf('Cy/0_type',type,
'Cy/0': $real ).
thf('Cz/0_type',type,
'Cz/0': $real ).
thf(p_qustion,conjecture,
( 'find/1' @ '3d.Shape'
@ ^ [V_Gs: '3d.Shape'] :
? [V_eps: '3d.Shape',V_A: '3d.Point',V_B: '3d.Point',V_C: '3d.Point'] :
( ( V_eps = '3d.xy-plane/0' )
& ( V_A
= ( '3d.point/3' @ 'Ax/0' @ 'Ay/0' @ 'Az/0' ) )
& ( V_B
= ( '3d.point/3' @ 'Bx/0' @ 'By/0' @ 'Bz/0' ) )
& ( V_C
= ( '3d.point/3' @ 'Cx/0' @ 'Cy/0' @ 'Cz/0' ) )
& ( $less @ 0.0 @ 'Az/0' )
& ( $less @ 0.0 @ 'Bz/0' )
& ( $less @ 0.0 @ 'Cz/0' )
& ~ ( '3d.colinear/3' @ V_A @ V_B @ V_C )
& ~ ( '3d.parallel/2' @ ( '3d.plane1/3' @ V_A @ V_B @ V_C ) @ V_eps )
& ( V_Gs
= ( '3d.shape-of-cpfun/1'
@ ^ [V_G: '3d.Point'] :
? [V_A2: '3d.Point',V_B2: '3d.Point',V_C2: '3d.Point',V_L: '3d.Point',V_M: '3d.Point',V_N: '3d.Point'] :
( ( '3d.on/2' @ V_A2 @ V_eps )
& ( '3d.on/2' @ V_B2 @ V_eps )
& ( '3d.on/2' @ V_C2 @ V_eps )
& ( ( '3d.midpoint-of/2' @ V_A @ V_A2 )
= V_L )
& ( ( '3d.midpoint-of/2' @ V_B @ V_B2 )
= V_M )
& ( ( '3d.midpoint-of/2' @ V_C @ V_C2 )
= V_N )
& ( '3d.is-triangle/3' @ V_L @ V_M @ V_N )
& ( V_G
= ( '3d.centroid-of/1' @ ( '3d.triangle/3' @ V_L @ V_M @ V_N ) ) ) ) ) ) ) ) ).
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