TPTP Problem File: GEO382^1.p
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% File : GEO382^1 : TPTP v9.0.0. Released v7.0.0.
% Domain : Geometry (Quadrangles)
% Problem : International Mathematical Olympiad, 1960, Problem 7
% Version : [Mat16] axioms : Especial.
% English : An isosceles trapezoid with bases a and c and altitude h is
% given. (a) On the axis of symmetry of this trapezoid, find all
% points P such that both legs of the trapezoid subtend right angles
% at P. (b) Calculate the distance of P from either base.
% (c) Determine under what conditions such points P actually exist.
% (Discuss various cases that might arise.)
% Refs : [Mat16] Matsuzaki (2016), Email to Geoff Sutcliffe
% : [MI+16] Matsuzaki et al. (2016), Race against the Teens - Benc
% Source : [Mat16]
% Names : IMO-1960-7.p [Mat16]
% Status : Theorem
% Rating : ? v7.0.0
% Syntax : Number of formulae : 3488 ( 727 unt;1202 typ; 0 def)
% Number of atoms : 6706 (2214 equ; 0 cnn)
% Maximal formula atoms : 22 ( 2 avg)
% Number of connectives : 39641 ( 104 ~; 233 |;1181 &;35997 @)
% (1095 <=>;1031 =>; 0 <=; 0 <~>)
% Maximal formula depth : 33 ( 8 avg)
% Number arithmetic : 4483 ( 374 atm;1211 fun; 962 num;1936 var)
% Number of types : 40 ( 36 usr; 3 ari)
% Number of type conns : 2408 (2408 >; 0 *; 0 +; 0 <<)
% Number of symbols : 1216 (1173 usr; 70 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: 5; Author: Jumma Kudo;
% Generated: 2014-12-04
% : Answer
% ^ [V_P_dot_0: '2d.Point'] :
% ? [V_Px: $real,V_Py: $real] :
% ( ( ( $less @ 0.0 @ 'h/0' )
% & ( $less @ 0.0 @ 'a/0' )
% & ( $less @ 0.0 @ 'b/0' )
% & ( $greatereq @ ( '^/2' @ 'h/0' @ 2.0 ) @ ( $product @ 'a/0' @ 'b/0' ) )
% & ( V_Px = 0.0 )
% & ( V_Py
% = ( $difference @ ( $quotient @ 'h/0' @ 2.0 ) @ ( $quotient @ ( 'sqrt/1' @ ( $difference @ ( '^/2' @ 'h/0' @ 2.0 ) @ ( $product @ 'a/0' @ 'b/0' ) ) ) @ 2.0 ) ) )
% & ( V_P_dot_0
% = ( '2d.point/2' @ V_Px @ V_Py ) ) )
% | ( ( $less @ 0.0 @ 'h/0' )
% & ( $less @ 0.0 @ 'a/0' )
% & ( $less @ 0.0 @ 'b/0' )
% & ( $greatereq @ ( '^/2' @ 'h/0' @ 2.0 ) @ ( $product @ 'a/0' @ 'b/0' ) )
% & ( V_Px = 0.0 )
% & ( V_Py
% = ( $sum @ ( $quotient @ 'h/0' @ 2.0 ) @ ( $quotient @ ( 'sqrt/1' @ ( $difference @ ( '^/2' @ 'h/0' @ 2.0 ) @ ( $product @ 'a/0' @ 'b/0' ) ) ) @ 2.0 ) ) )
% & ( V_P_dot_0
% = ( '2d.point/2' @ V_Px @ V_Py ) ) ) ) )
%------------------------------------------------------------------------------
include('Axioms/MAT001^0.ax').
%------------------------------------------------------------------------------
thf('a/0_type',type,
'a/0': $real ).
thf('b/0_type',type,
'b/0': $real ).
thf('h/0_type',type,
'h/0': $real ).
thf(p1_qustion,conjecture,
( 'find/1' @ '2d.Point'
@ ^ [V_P: '2d.Point'] :
? [V_A: '2d.Point',V_B: '2d.Point',V_C: '2d.Point',V_D: '2d.Point'] :
( ( $less @ 0.0 @ 'h/0' )
& ( $less @ 0.0 @ 'a/0' )
& ( $less @ 0.0 @ 'b/0' )
& ( V_A
= ( '2d.point/2' @ ( $uminus @ ( $quotient @ 'b/0' @ 2.0 ) ) @ 0.0 ) )
& ( V_B
= ( '2d.point/2' @ ( $quotient @ 'b/0' @ 2.0 ) @ 0.0 ) )
& ( V_C
= ( '2d.point/2' @ ( $quotient @ 'a/0' @ 2.0 ) @ 'h/0' ) )
& ( V_D
= ( '2d.point/2' @ ( $uminus @ ( $quotient @ 'a/0' @ 2.0 ) ) @ 'h/0' ) )
& ( ( '2d.rad-of-angle/1' @ ( '2d.angle/3' @ V_B @ V_P @ V_C ) )
= ( $quotient @ 'Pi/0' @ 2.0 ) )
& ( ( '2d.rad-of-angle/1' @ ( '2d.angle/3' @ V_A @ V_P @ V_D ) )
= ( $quotient @ 'Pi/0' @ 2.0 ) )
& ( '2d.on/2' @ V_P @ '2d.y-axis/0' ) ) ) ).
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