TPTP Problem File: GEO416^1.p
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% File : GEO416^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Geometry (Polygons)
% Problem : International Mathematical Olympiad, 1995, Problem 5
% Version : [Mat16] axioms : Especial.
% English : Let ABCDEF be a convex hexagon with AB = BC = CD and DE = EF =
% FA, such that angle BCD = angle EFA = pi/3. Suppose G and H
% are points in the interior of the hexagon such that angle AGB =
% angle DHE = 2 pi/3. Prove that AG + GB + GH + DH + HE >= CF.
% Refs : [Mat16] Matsuzaki (2016), Email to Geoff Sutcliffe
% : [MI+16] Matsuzaki et al. (2016), Race against the Teens - Benc
% Source : [Mat16]
% Names : IMO-1995-5.p [Mat16]
% Status : Theorem
% Rating : ? v7.0.0
% Syntax : Number of formulae : 3485 ( 711 unt;1199 typ; 0 def)
% Number of atoms : 8228 (2216 equ; 0 cnn)
% Maximal formula atoms : 42 ( 3 avg)
% Number of connectives : 39734 ( 104 ~; 233 |;1182 &;36088 @)
% (1095 <=>;1032 =>; 0 <=; 0 <~>)
% Maximal formula depth : 33 ( 8 avg)
% Number arithmetic : 4478 ( 372 atm;1213 fun; 957 num;1936 var)
% Number of types : 40 ( 36 usr; 3 ari)
% Number of type conns : 2408 (2408 >; 0 *; 0 +; 0 <<)
% Number of symbols : 1215 (1172 usr; 69 con; 0-9 aty)
% Number of variables : 8063 ( 405 ^;7093 !; 429 ?;8063 :)
% ( 136 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_ARI
% Comments : Theory: RCF; Score: 7; Author: Yiyang Zhan;
% Generated: 2014-12-10
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include('Axioms/MAT001^0.ax').
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thf(p,conjecture,
! [V_A: '2d.Point',V_B: '2d.Point',V_C: '2d.Point',V_D: '2d.Point',V_E: '2d.Point',V_F: '2d.Point',V_G: '2d.Point',V_H: '2d.Point'] :
( ( ( '2d.is-convex-shape/1' @ ( '2d.polygon/1' @ ( 'cons/2' @ '2d.Point' @ V_A @ ( 'cons/2' @ '2d.Point' @ V_B @ ( 'cons/2' @ '2d.Point' @ V_C @ ( 'cons/2' @ '2d.Point' @ V_D @ ( 'cons/2' @ '2d.Point' @ V_E @ ( 'cons/2' @ '2d.Point' @ V_F @ ( 'nil/0' @ '2d.Point' ) ) ) ) ) ) ) ) )
& ( ( '2d.distance/2' @ V_A @ V_B )
= ( '2d.distance/2' @ V_B @ V_C ) )
& ( ( '2d.distance/2' @ V_B @ V_C )
= ( '2d.distance/2' @ V_C @ V_D ) )
& ( ( '2d.distance/2' @ V_D @ V_E )
= ( '2d.distance/2' @ V_E @ V_F ) )
& ( ( '2d.distance/2' @ V_E @ V_F )
= ( '2d.distance/2' @ V_F @ V_A ) )
& ( ( '2d.rad-of-angle/1' @ ( '2d.angle/3' @ V_B @ V_C @ V_D ) )
= ( $quotient @ 'Pi/0' @ 3.0 ) )
& ( ( '2d.rad-of-angle/1' @ ( '2d.angle/3' @ V_E @ V_F @ V_A ) )
= ( $quotient @ 'Pi/0' @ 3.0 ) )
& ( '2d.point-inside-of/2' @ V_G @ ( '2d.polygon/1' @ ( 'cons/2' @ '2d.Point' @ V_A @ ( 'cons/2' @ '2d.Point' @ V_B @ ( 'cons/2' @ '2d.Point' @ V_C @ ( 'cons/2' @ '2d.Point' @ V_D @ ( 'cons/2' @ '2d.Point' @ V_E @ ( 'cons/2' @ '2d.Point' @ V_F @ ( 'nil/0' @ '2d.Point' ) ) ) ) ) ) ) ) )
& ( '2d.point-inside-of/2' @ V_H @ ( '2d.polygon/1' @ ( 'cons/2' @ '2d.Point' @ V_A @ ( 'cons/2' @ '2d.Point' @ V_B @ ( 'cons/2' @ '2d.Point' @ V_C @ ( 'cons/2' @ '2d.Point' @ V_D @ ( 'cons/2' @ '2d.Point' @ V_E @ ( 'cons/2' @ '2d.Point' @ V_F @ ( 'nil/0' @ '2d.Point' ) ) ) ) ) ) ) ) )
& ( ( '2d.rad-of-angle/1' @ ( '2d.angle/3' @ V_A @ V_G @ V_B ) )
= ( $quotient @ ( $product @ 2.0 @ 'Pi/0' ) @ 3.0 ) )
& ( ( '2d.rad-of-angle/1' @ ( '2d.angle/3' @ V_D @ V_H @ V_E ) )
= ( $quotient @ ( $product @ 2.0 @ 'Pi/0' ) @ 3.0 ) ) )
=> ( $greatereq @ ( $sum @ ( '2d.distance/2' @ V_A @ V_G ) @ ( $sum @ ( '2d.distance/2' @ V_G @ V_B ) @ ( $sum @ ( '2d.distance/2' @ V_G @ V_H ) @ ( $sum @ ( '2d.distance/2' @ V_D @ V_H ) @ ( '2d.distance/2' @ V_H @ V_E ) ) ) ) ) @ ( '2d.distance/2' @ V_C @ V_F ) ) ) ).
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