TPTP Problem File: GEO379^1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : GEO379^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Geometry (Lines and planes in 3D space)
% Problem : International Mathematical Olympiad, 1959, Problem 6
% Version : [Mat16] axioms : Especial.
% English : Two planes, P and Q, intersect along the line p. The point A is
% given in the plane P, and the point C in the plane Q; neither of
% these points lies on the straight line p. Construct an isosceles
% trapezoid ABCD (with AB parallel to CD) in which a circle can be
% inscribed, and with vertices B and D lying in the planes P and Q
% respectively.
% Refs : [Mat16] Matsuzaki (2016), Email to Geoff Sutcliffe
% : [MI+16] Matsuzaki et al. (2016), Race against the Teens - Benc
% Source : [Mat16]
% Names : IMO-1959-6.p [Mat16]
% Status : Theorem
% Rating : ? v7.0.0
% Syntax : Number of formulae : 3499 ( 710 unt;1213 typ; 0 def)
% Number of atoms : 8712 (2215 equ; 0 cnn)
% Maximal formula atoms : 40 ( 3 avg)
% Number of connectives : 39674 ( 106 ~; 233 |;1189 &;36020 @)
% (1095 <=>;1031 =>; 0 <=; 0 <~>)
% Maximal formula depth : 33 ( 8 avg)
% Number arithmetic : 4461 ( 371 atm;1203 fun; 951 num;1936 var)
% Number of types : 40 ( 36 usr; 3 ari)
% Number of type conns : 2408 (2408 >; 0 *; 0 +; 0 <<)
% Number of symbols : 1237 (1194 usr; 91 con; 0-9 aty)
% Number of variables : 8064 ( 406 ^;7085 !; 437 ?;8064 :)
% ( 136 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_ARI
% Comments : Theory: RCF; Score: 7; Author: Jumma Kudo;
% Generated: 2014-12-16
% : Answer
% ^ [V_BD_dot_0: ( 'ListOf' @ '3d.Point' )] :
% ? [V_a: '3d.Shape',V_c: '3d.Shape',V_A_dot_0: '3d.Point',V_AA: '3d.Point',V_C_dot_0: '3d.Point',V_p_dot_0: '3d.Shape',V_alpha_dot_0: '3d.Shape',V_beta_dot_0: '3d.Shape',V_B_dot_0: '3d.Point',V_D_dot_0: '3d.Point'] :
% ( ( V_alpha_dot_0
% = ( '3d.plane/4' @ 'a1/0' @ 'b1/0' @ 'c1/0' @ 'd1/0' ) )
% & ( V_beta_dot_0
% = ( '3d.plane/4' @ 'a2/0' @ 'b2/0' @ 'c2/0' @ 'd2/0' ) )
% & ( V_A_dot_0
% = ( '3d.point/3' @ 'Ax/0' @ 'Ay/0' @ 'Az/0' ) )
% & ( V_C_dot_0
% = ( '3d.point/3' @ 'Cx/0' @ 'Cy/0' @ 'Cz/0' ) )
% & ( '3d.on/2' @ V_A_dot_0 @ V_alpha_dot_0 )
% & ( '3d.on/2' @ V_C_dot_0 @ V_beta_dot_0 )
% & ( '3d.line-type/1' @ V_p_dot_0 )
% & ( ( '3d.intersection/2' @ V_alpha_dot_0 @ V_beta_dot_0 )
% = V_p_dot_0 )
% & ( '3d.line-type/1' @ V_a )
% & ( '3d.line-type/1' @ V_c )
% & ( '3d.on/2' @ V_A_dot_0 @ V_a )
% & ( '3d.on/2' @ V_C_dot_0 @ V_c )
% & ( '3d.parallel/2' @ V_a @ V_p_dot_0 )
% & ( '3d.parallel/2' @ V_c @ V_p_dot_0 )
% & ( '3d.on/2' @ V_AA @ V_c )
% & ( '3d.perpendicular/2' @ ( '3d.line/2' @ V_A_dot_0 @ V_AA ) @ V_c )
% & ( ( '3d.length-of/1' @ ( '3d.seg/2' @ V_AA @ V_C_dot_0 ) )
% = ( '3d.length-of/1' @ ( '3d.seg/2' @ V_C_dot_0 @ V_B_dot_0 ) ) )
% & ( ( '3d.length-of/1' @ ( '3d.seg/2' @ V_AA @ V_C_dot_0 ) )
% = ( '3d.length-of/1' @ ( '3d.seg/2' @ V_A_dot_0 @ V_D_dot_0 ) ) )
% & ( '3d.on/2' @ V_B_dot_0 @ V_a )
% & ( '3d.on/2' @ V_D_dot_0 @ V_c )
% & ( V_BD_dot_0
% = ( 'cons/2' @ '3d.Point' @ V_B_dot_0 @ ( 'cons/2' @ '3d.Point' @ V_D_dot_0 @ ( 'nil/0' @ '3d.Point' ) ) ) )
% & ( '3d.vec-opp-direction/2' @ ( '3d.vec/2' @ V_A_dot_0 @ V_B_dot_0 ) @ ( '3d.vec/2' @ V_C_dot_0 @ V_D_dot_0 ) ) ) )
%------------------------------------------------------------------------------
include('Axioms/MAT001^0.ax').
%------------------------------------------------------------------------------
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('Cx/0_type',type,
'Cx/0': $real ).
thf('Cy/0_type',type,
'Cy/0': $real ).
thf('Cz/0_type',type,
'Cz/0': $real ).
thf('a1/0_type',type,
'a1/0': $real ).
thf('a2/0_type',type,
'a2/0': $real ).
thf('b1/0_type',type,
'b1/0': $real ).
thf('b2/0_type',type,
'b2/0': $real ).
thf('c1/0_type',type,
'c1/0': $real ).
thf('c2/0_type',type,
'c2/0': $real ).
thf('d1/0_type',type,
'd1/0': $real ).
thf('d2/0_type',type,
'd2/0': $real ).
thf(p_qustion,conjecture,
( 'find/1' @ ( 'ListOf' @ '3d.Point' )
@ ^ [V_BD: 'ListOf' @ '3d.Point'] :
? [V_A: '3d.Point',V_C: '3d.Point',V_B: '3d.Point',V_D: '3d.Point',V_p: '3d.Shape',V_alpha: '3d.Shape',V_beta: '3d.Shape',V_K: '3d.Shape'] :
( ( V_alpha
= ( '3d.plane/4' @ 'a1/0' @ 'b1/0' @ 'c1/0' @ 'd1/0' ) )
& ( V_beta
= ( '3d.plane/4' @ 'a2/0' @ 'b2/0' @ 'c2/0' @ 'd2/0' ) )
& ( V_A
= ( '3d.point/3' @ 'Ax/0' @ 'Ay/0' @ 'Az/0' ) )
& ( V_C
= ( '3d.point/3' @ 'Cx/0' @ 'Cy/0' @ 'Cz/0' ) )
& ( '3d.on/2' @ V_A @ V_alpha )
& ( '3d.on/2' @ V_C @ V_beta )
& ( '3d.line-type/1' @ V_p )
& ( ( '3d.intersection/2' @ V_alpha @ V_beta )
= V_p )
& ~ ( '3d.on/2' @ V_A @ V_p )
& ~ ( '3d.on/2' @ V_C @ V_p )
& ( '3d.on/2' @ V_B @ V_alpha )
& ( '3d.on/2' @ V_D @ V_beta )
& ( '3d.is-trapezoid/4' @ V_A @ V_B @ V_C @ V_D )
& ( '3d.parallel/2' @ ( '3d.line/2' @ V_A @ V_B ) @ ( '3d.line/2' @ V_C @ V_D ) )
& ( ( '3d.length-of/1' @ ( '3d.seg/2' @ V_A @ V_C ) )
= ( '3d.length-of/1' @ ( '3d.seg/2' @ V_B @ V_D ) ) )
& ( '3d.circle-type/1' @ V_K )
& ( '3d.is-inscribed-in/2' @ V_K @ ( '3d.square/4' @ V_A @ V_B @ V_C @ V_D ) )
& ( V_BD
= ( 'cons/2' @ '3d.Point' @ V_B @ ( 'cons/2' @ '3d.Point' @ V_D @ ( 'nil/0' @ '3d.Point' ) ) ) ) ) ) ).
%------------------------------------------------------------------------------