True, false or neither true nor false statements

M. Pilar Vélez, Feb 16, 2018

This GeoGebraBook contains some examples about the new GeoGebra features for automated reasoning in Elementary Geometry. The background of these new features is based in algebraic geometry methods performed internally by GeoGebra. We can discover and/or prove relations between objects in a GeoGebra construction. In this context, statements could be: [list] [*]"Always true": true for all instances of the geometric construction. [*]"Generally true": true except if degenerate. [*]"False in general": false except for some degenarate case. [*]"True on parts, false on parts": a particular issue, namely, the case of statements that are true on non degenerate parts of the construction and false in some others, i.e. neithr true nor false. [/list]

Table of Contents

  1. Always true and generally true
    1. Chou 179
  2. False in general
    1. Triangle altitude and base midpoint
  3. True on parts, false on parts
    1. A simple example on "true on parts, false on parts"
    2. Diagonals of a rhombus
    3. Equilateral triangles on the sides of a triangle
    4. Squares on the sides of a triangle
    5. Regular pentagon erected internally on a side of a regular decagon

1.

Always true and generally true

  • 1. Chou 179

1.1.

Chou 179

Consider a triangle ABC and construct its orthocenter D (intersection of altitudes). Then, create a circumcircle d through points B, C and D and two lines through D parallels to the sides b and c respectivelly. Take E and F to be the intersection points of these parallel lines with circle d and consider triangle DEF.[br]Ask about relations between segments AB and DE. [br][br]This example appears as example 179 in: Chou, S.C.: Mechanical geometry theorem proving, Mathematics and its Applications, vol. 41. D. Reidel Publishing Co., Dordrecht (1988)[br]
What about segments AC and DF?[br]And... What about segments BC and EF?
– Zoltán Kovács, M. Pilar Vélez

2.

False in general

  • 1. Triangle altitude and base midpoint

2.1.

Triangle altitude and base midpoint

Consider a triangle ABC and its altitude from B. Then construct the midpoint D of AC. Check if the altitude intersects side AC on D.
Ask about the relation between D and f.
– Zoltán Kovács, M. Pilar Vélez

3.

True on parts, false on parts

  • 1. A simple example on "true on parts, false on parts"
  • 2. Diagonals of a rhombus
  • 3. Equilateral triangles on the sides of a triangle
  • 4. Squares on the sides of a triangle
  • 5. Regular pentagon erected internally on a side of a regular decagon

3.1.

A simple example on "true on parts, false on parts"

Draw two points A and B, and a circle c centered at A and passing through B. Now trace a line f passing through A and B and consider the intersection points, C and D, of the circle c with the line f . Ask about the equality between B and C.[br][br]This is a very simple example of automated reasoning in GeoGebra that shows a conflict and explains why some statements are neither true nor false.
Use the Relation tool to ask about B and C
Are you expected some different answer?
Why is this statement considered as "true on parts, false on parts"?[br]Think about how points C and D are constructed.
– Zoltán Kovács, M. Pilar Vélez

3.2.

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3.4.

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3.5.

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