TPTP Problem File: GEO417^1.p
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% File : GEO417^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Geometry (Polygons)
% Problem : International Mathematical Olympiad, 1996, Problem 5
% Version : [Mat16] axioms : Especial.
% English : Let ABCDEF be a convex hexagon such that AB is parallel to DE,
% BC is parallel to EF, and CD is parallel to FA. Let R_A, R_C,
% R_E denote the circumradii of triangles FAB, BCD, DEF,
% respectively, and let P denote the perimeter of the hexagon.
% Prove that R_A + R_C + R_E >= P/2.
% Refs : [Mat16] Matsuzaki (2016), Email to Geoff Sutcliffe
% : [MI+16] Matsuzaki et al. (2016), Race against the Teens - Benc
% Source : [Mat16]
% Names : IMO-1996-5.p [Mat16]
% Status : Theorem
% Rating : ? v7.0.0
% Syntax : Number of formulae : 3485 ( 711 unt;1199 typ; 0 def)
% Number of atoms : 8661 (2212 equ; 0 cnn)
% Maximal formula atoms : 40 ( 3 avg)
% Number of connectives : 39702 ( 104 ~; 233 |;1185 &;36053 @)
% (1095 <=>;1032 =>; 0 <=; 0 <~>)
% Maximal formula depth : 33 ( 8 avg)
% Number arithmetic : 4475 ( 372 atm;1211 fun; 952 num;1940 var)
% Number of types : 40 ( 36 usr; 3 ari)
% Number of type conns : 2408 (2408 >; 0 *; 0 +; 0 <<)
% Number of symbols : 1218 (1175 usr; 72 con; 0-9 aty)
% Number of variables : 8068 ( 405 ^;7098 !; 429 ?;8068 :)
% ( 136 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_ARI
% Comments : Theory: RCF; Score: 7; Author: Jumma Kudo;
% Generated: 2014-11-06
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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_K1: '2d.Shape',V_K2: '2d.Shape',V_K3: '2d.Shape',V_RA: $real,V_RC: $real,V_RE: $real,V_P: $real] :
( ( ( '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.parallel/2' @ ( '2d.line/2' @ V_A @ V_B ) @ ( '2d.line/2' @ V_D @ V_E ) )
& ( '2d.parallel/2' @ ( '2d.line/2' @ V_B @ V_C ) @ ( '2d.line/2' @ V_E @ V_F ) )
& ( '2d.parallel/2' @ ( '2d.line/2' @ V_C @ V_D ) @ ( '2d.line/2' @ V_A @ V_F ) )
& ( '2d.circle-type/1' @ V_K1 )
& ( '2d.circle-type/1' @ V_K2 )
& ( '2d.circle-type/1' @ V_K3 )
& ( '2d.is-inscribed-in/2' @ ( '2d.triangle/3' @ V_F @ V_A @ V_B ) @ V_K1 )
& ( '2d.is-inscribed-in/2' @ ( '2d.triangle/3' @ V_B @ V_C @ V_D ) @ V_K2 )
& ( '2d.is-inscribed-in/2' @ ( '2d.triangle/3' @ V_D @ V_E @ V_F ) @ V_K3 )
& ( V_RA
= ( '2d.radius-of/1' @ V_K1 ) )
& ( V_RC
= ( '2d.radius-of/1' @ V_K2 ) )
& ( V_RE
= ( '2d.radius-of/1' @ V_K3 ) )
& ( V_P
= ( $sum @ ( '2d.length-of/1' @ ( '2d.seg/2' @ V_A @ V_B ) ) @ ( $sum @ ( '2d.length-of/1' @ ( '2d.seg/2' @ V_B @ V_C ) ) @ ( $sum @ ( '2d.length-of/1' @ ( '2d.seg/2' @ V_C @ V_D ) ) @ ( $sum @ ( '2d.length-of/1' @ ( '2d.seg/2' @ V_D @ V_E ) ) @ ( $sum @ ( '2d.length-of/1' @ ( '2d.seg/2' @ V_E @ V_F ) ) @ ( '2d.length-of/1' @ ( '2d.seg/2' @ V_F @ V_A ) ) ) ) ) ) ) ) )
=> ( $greatereq @ ( $sum @ V_RA @ ( $sum @ V_RC @ V_RE ) ) @ ( $quotient @ V_P @ 2.0 ) ) ) ).
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