The Visual Effects of All Assertions .

Atomic Predicates:

P1 collinear A B C A, B, and C are collinear
P2 parallel A B C D AB and CD are parallel
P3 perpendicular A B C D AB and CD are perpendicular
P4 eq-distance A B C D |AB| = |CD|
P5 cyclic A B C D A, B, C and D are cyclic
P6 eq-angle [A B C D] [A1 B1 C1 D1] full-angles ∠[ABCD] = ∠[A1B1C1D1]
P7 eq-angle A B C A1 B1 C1 traditional angles ∠ABC = ∠A1B1C1
P8 congruent A B C D E F triangle ABC is congruent to triangle DEF
P9 similiar A B C D E F triangle ABC is similiar to triangle DEF
P10 eq-ratio [A B C D] [A1 B1 C1 D1] AB / CD = A1B1 / C1D1

Compounded Predicates

P11 midpoint M A B M is the midpoint of AB, i.e., A, M, B are collinear and AM = MB
P12 lratio C A B n m A, B, C are collinear and AC / n = CB / m
P13 pratio C D A B n m AB and CD are parallel and CD / n = AB / m
P14 tratio C D A B n m AB and CD are perpendicular and CD / AB = n / m ;where n and m are two positive integers.

Compounded Predicates (Construction-Related)

P15 triangle A B C A, B, and C are not collinear
P16 R-triangle C A B AC ⊥ BC and A, B, and C are not collinear
P17 iso-triangle C A B CA = CB and A, B, C are not collinear
P18 iso-right triangle C A B BC ⊥ AC, BC = AC, A, B, and C are not collinear
P19 foot D C A B CD ⊥ AB, A, C and D are collinear, and AB is non-isotropic
P20 equi-triangle A B C AB = BC = CA, ∠ABC = ∠CAB∠BCA = 60◦. A, B, C are not collinear and are in the counter-clockwise orientation
P21 trapezoid A B C D AB and CD are parallel & AB and CD have the same direction, which is equivalent to AB and CD are parallel, where AB denotes the oriented segment of AB. Also we need a nondegenerate condition: A, B, and C are not collinear.
P22 parallelogram A B C D AB // CD, BC // AD, and A, B, C are not collinear
P23 rectangle A B C D Parallelogram A B C D and DA ⊥ BA
P24 square A B C D Rectangle A B C D and AB = AD

Inequality Predicates:

P25 between A B C A, B, and C are collinear, B != C, and A is between B and C
P26 angle-inside D A B C D is inside (traditional) angle ∠ ABC
P27 angle-outside D A B C D is outside ∠ ABC
P28 triangle-inside D A B C D is inside triangle ABC
P29 paral-inside E A B C D E is inside parallelogram ABCD
P30 opposite-side C D A B C and D are on the opposite sides of line AB, and A != B
P31 same-side C D A B points C and D are on the same side of line AB , and A != B
P32 AB < DC segment AB is less than segment DC
P33 ∠ ABC < ∠ CDE ∠ ABC < ∠ CDE
P34 The (n + 1)-polygon A0A1A2 . . . An is convex (n ≥ 3). points Ai and Ai+3 are on the same side of line Ai+1Ai+2, where indexes are understood mod n + 1.

 

The Visual Effects of All Predicates.

P1. collinear A, B, C
Points A, B, C and line ABC are all highlighted.

P2. parallel A, B, C, D
Two endless lines are drawn to overlap linesAB and CD. The points on the two lines are highlighted also.

P3. perpendicular A, B, C, D
Segments AB and CD are highlighted and a foot mark is drawn on the diagram.

P4. eq-distance A, B, C, D
AB and CD are highlighted and a copy of AB is moved on the fly to overlap CD.

P5. cyclic A, B, C, D
Draw an auxiliary circle, and highlight the points A, B, C, D.

P6. eq-angle [A B C D] [A1 B1 C1 D1]
full-angles ∠[ABCD] = ∠[A1B1C1D1]

P7. eq-angle A B C A1 B1 C1
traditional angles ∠ABC = ∠A1B1C1

 

P8. cogruent A, B, C, D, E, F
The program moves a copy of color-filled triangle ABC on the fly by a rotation followed by a translation, possibly with a flip (reflection) if the orientations of the two triangles are opposite, and drops it to the position of the triangle DEF, at last blinks the corresponding sides of the two triangles alternately.

P9. similiar A, B, C, D, E, F
A copy of triangle ABC first scales to the size of triangle DEF on the fly so that the two triangles can fit. The other visual effects are the same as those for two congruent triangles mentioned above.

P10. eq-ratio [A B C D] [A1 B1 C1 D1]
AB / CD = A1B1 / C1D1

P11. midpoint M A B
M is the midpoint of AB, i.e., A, M, B are collinear and AM = MB. The Effects is same as P4.

 

P12. lratio C A B n m
P13. pratio C D A B n m
P14. tratio C D A B n m
Two related segements are highlighted.

P15. triangle A B C
triangle ABC are filled with color.

P16. R-triangle C A B
Combination of P3 and P13.

P17. iso-triangle C A B
Combination of P3 and P13.

P18. iso-right triangle C A B
Combination of P3 and P13.

 

P19. foot D C A B
Same as P16.

P20. equi-triangle A B C
triangle ABC is filled with default color and three equal mark are put on each segments.

 

P21. trapezoid A B C D
P22. parallelogram A B C D
P23. rectangle A B C D
P24. square A B C D

 

P25. between A B C
Combination.

 

P26. angle-inside D A B C
Combination of P3 and P13.

P27. angle-outside D A B C
Combination of P3 and P13.

P28. triangle-inside D A B C
The triangle ABC is filled with default color and the point D is drawn with a slight longer radius.

P29. paral-inside E A B C D
the parallelogram ABCD is filled with default color. Point E is drawn with a slight longer radius to make it stand out.

 

P30. opposite-side C D A B
An endless line is drawn to overlap segment AB.

 

P31. same-side C D A B
An endless line is drawn to overlap segment AB. Two arrows start from this line and point to C and D respectively are drawn.

P32. AB < DC
The segments AB and CD are highlighted respectively.

P33. ∠ ABC < ∠ CDE
Draw two traditional angle with value and blink.

P34. The (n + 1)-polygon A0A1A2 . . . An is convex (n ≥ 3).
We just fill the polygon with default color and blink its.

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