TPTP Problem File: GEO389^1.p
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% File : GEO389^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Geometry (Spheres and other surfaces)
% Problem : International Mathematical Olympiad, 1963, Problem 2
% Version : [Mat16] axioms : Especial.
% English : Point A and segment BC are given. Determine the locus of points
% in space which are vertices of right angles with one side passing
% through A, and the other side intersecting the segment BC.
% Refs : [Mat16] Matsuzaki (2016), Email to Geoff Sutcliffe
% : [MI+16] Matsuzaki et al. (2016), Race against the Teens - Benc
% Source : [Mat16]
% Names : IMO-1963-2.p [Mat16]
% Status : Theorem
% Rating : ? v7.0.0
% Syntax : Number of formulae : 3494 ( 727 unt;1208 typ; 0 def)
% Number of atoms : 6802 (2214 equ; 0 cnn)
% Maximal formula atoms : 22 ( 2 avg)
% Number of connectives : 39627 ( 107 ~; 233 |;1179 &;35982 @)
% (1095 <=>;1031 =>; 0 <=; 0 <~>)
% Maximal formula depth : 33 ( 8 avg)
% Number arithmetic : 4463 ( 371 atm;1204 fun; 952 num;1936 var)
% Number of types : 40 ( 36 usr; 3 ari)
% Number of type conns : 2408 (2408 >; 0 *; 0 +; 0 <<)
% Number of symbols : 1223 (1179 usr; 77 con; 0-9 aty)
% Number of variables : 8060 ( 406 ^;7085 !; 433 ?;8060 :)
% ( 136 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_ARI
% Comments : Theory: RCF; Score: 7; Author: Jumma Kudo;
% Generated: 2014-12-17
% : Answer
% ^ [V_P_dot_0: '3d.Point'] :
% ? [V_A_dot_0: '3d.Point',V_B_dot_0: '3d.Point',V_C_dot_0: '3d.Point',V_rB2: $real,V_rC2: $real] :
% ( ( V_A_dot_0
% = ( '3d.point/3' @ 'Ax/0' @ 'Ay/0' @ 'Az/0' ) )
% & ( V_B_dot_0
% = ( '3d.point/3' @ 'Bx/0' @ 'By/0' @ 'Bz/0' ) )
% & ( V_C_dot_0
% = ( '3d.point/3' @ 'Cx/0' @ 'Cy/0' @ 'Cz/0' ) )
% & ( V_B_dot_0 != V_C_dot_0 )
% & ~ ( '3d.on/2' @ V_A_dot_0 @ ( '3d.seg/2' @ V_B_dot_0 @ V_C_dot_0 ) )
% & ( V_rB2
% = ( $quotient @ ( '3d.distance/2' @ V_A_dot_0 @ V_B_dot_0 ) @ 2.0 ) )
% & ( V_rC2
% = ( $quotient @ ( '3d.distance/2' @ V_A_dot_0 @ V_C_dot_0 ) @ 2.0 ) )
% & ( V_P_dot_0 != V_A_dot_0 )
% & ( ( $lesseq @ ( '3d.distance/2' @ V_P_dot_0 @ ( '3d.midpoint-of/2' @ V_A_dot_0 @ V_B_dot_0 ) ) @ V_rB2 )
% | ( $lesseq @ ( '3d.distance/2' @ V_P_dot_0 @ ( '3d.midpoint-of/2' @ V_A_dot_0 @ V_C_dot_0 ) ) @ V_rC2 ) )
% & ( ( $greatereq @ ( '3d.distance/2' @ V_P_dot_0 @ ( '3d.midpoint-of/2' @ V_A_dot_0 @ V_B_dot_0 ) ) @ V_rB2 )
% | ( $greatereq @ ( '3d.distance/2' @ V_P_dot_0 @ ( '3d.midpoint-of/2' @ V_A_dot_0 @ V_C_dot_0 ) ) @ V_rC2 ) ) ) )
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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.Point'
@ ^ [V_P: '3d.Point'] :
? [V_A: '3d.Point',V_B: '3d.Point',V_C: '3d.Point',V_X: '3d.Point'] :
( ( 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' ) )
& ( V_B != V_C )
& ~ ( '3d.on/2' @ V_A @ ( '3d.seg/2' @ V_B @ V_C ) )
& ( V_X != V_A )
& ( '3d.on/2' @ V_X @ ( '3d.seg/2' @ V_B @ V_C ) )
& ( ( '3d.rad-of-angle/1' @ ( '3d.angle/3' @ V_A @ V_P @ V_X ) )
= ( $product @ 90.0 @ 'Degree/0' ) ) ) ) ).
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