How many triangles are there?

እነዚህ ትሪያንግሎች የላይኛው ጫፍ A ያላቸው እና መሰረታቸው በ EF ወይም CD ላይ ያሉ ናቸው።

• መሰረታቸው በ EF ላይ የሆኑት:
  1. $\triangle$ AGE
  2. $\triangle$ AGF
  3. $\triangle$ AEF (የ $\triangle$ AGE እና $\triangle$ AGF ጥምረት)
• መሰረታቸው በ CD ላይ የሆኑት: 4. $\triangle$ ACO 5. $\triangle$ ADO 6. $\triangle$ ACD (የ $\triangle$ ACO እና $\triangle$ ADO ጥምረት) በዚህ ክፍል ውስጥ ጠቅላላ = 6 ትሪያንግሎች አሉ።

እነዚህ ትሪያንግሎች የታችኛው ጫፍ B ያላቸው እና መሰረታቸው በ HI ወይም CD ላይ ያሉ ናቸው።

• መሰረታቸው በ HI ላይ የሆኑት: 7. $\triangle$ BHJ 8. $\triangle$ BIJ 9. $\triangle$ BHI (የ $\triangle$ BHJ እና $\triangle$ BIJ ጥምረት)
• መሰረታቸው በ CD ላይ የሆኑት: 10. $\triangle$ BCO 11. $\triangle$ BDO 12. $\triangle$ BCD (የ $\triangle$ BCO እና $\triangle$ BDO ጥምረት) በዚህ ክፍል ውስጥ ጠቅላላ = 6 ትሪያንግሎች አሉ።

እነዚህ ትሪያንግሎች ጫፋቸው C ወይም D ሲሆን መሰረታቸው በ AB ላይ ያሉ ናቸው።

• ጫፋቸው C የሆኑት: 13. $\triangle$ CGO (G, O በ AB ላይ ናቸው) 14. $\triangle$ CJO (J, O በ AB ላይ ናቸው) 15. $\triangle$ CGJ (G, J በ AB ላይ ናቸው) 16. $\triangle$ CGO + $\triangle$ CJO = $\triangle$ CGO (No, G, J, O are collinear) 17. $\triangle$ CJO (No, J, O are collinear) 18. $\triangle$ CGO (No, G, O are collinear) 19. $\triangle$ CJO (No, J, O are collinear) 20. $\triangle$ CGJ (No, G, J are collinear) 21. $\triangle$ CAO (No, A, O are collinear) 22. $\triangle$ CBO (No, B, O are collinear) 23. $\triangle$ CAO (No, A, O are collinear) 24. $\triangle$ CBO (No, B, O are collinear)

Let's re-evaluate the central triangles.
Triangles with base on the vertical line AB: 13. $\triangle$ CEG (No, E is on AC) 14. $\triangle$ CFG (No, F is on AD) 15. $\triangle$ CHJ (No, H is on BC) 16. $\triangle$ CIJ (No, I is on BD)

Let's list the triangles by their vertices again, to avoid double counting and missing any.
Points: A, B, C, D, E, F, G, H, I, J, O.

Triangles from Step 1 (top section):

1. $\triangle$ AGE
2. $\triangle$ AGF
3. $\triangle$ AEF
4. $\triangle$ ACO
5. $\triangle$ ADO
6. $\triangle$ ACD

Triangles from Step 2 (bottom section): 7. $\triangle$ BHJ 8. $\triangle$ BIJ 9. $\triangle$ BHI 10. $\triangle$ BCO 11. $\triangle$ BDO 12. $\triangle$ BCD

እነዚህ ትሪያንግሎች ጫፋቸው C ወይም D ሲሆን መሰረታቸው በ AB ላይ ያሉ ናቸው።

• ጫፋቸው C የሆኑት: 13. $\triangle$ CGO (C, G, O) 14. $\triangle$ CJO (C, J, O) 15. $\triangle$ CGJ (C, G, J) 16. $\triangle$ CAO (C, A, O) 17. $\triangle$ CBO (C, B, O) 18. $\triangle$ CAG (C, A, G) 19. $\triangle$ CAJ (C, A, J) 20. $\triangle$ CBJ (C, B, J) 21. $\triangle$ CBG (C, B, G)

Let's simplify the counting for the middle section.
Consider the left half (left of AB):

• $\triangle$ CGO
• $\triangle$ CJO
• $\triangle$ CGJ (CGO + CJO)
• $\triangle$ CAO (CGO + CAG)
• $\triangle$ CBO (CJO + CBJ)
• $\triangle$ CAG (No, G is on AB)
• $\triangle$ CAJ (No, J is on AB)
• $\triangle$ CBJ (No, J is on AB)
• $\triangle$ CBG (No, G is on AB)

Let's count triangles with base on the vertical line AB.
Triangles with base on GO: 13. $\triangle$ CGO 14. $\triangle$ DGO Triangles with base on JO: 15. $\triangle$ CJO 16. $\triangle$ DJO Triangles with base on GJ: 17. $\triangle$ CGJ (CGO + CJO) 18. $\triangle$ DGJ (DGO + DJO) Triangles with base on AO: 19. $\triangle$ CAO (CGO + CAG) 20. $\triangle$ DAO (DGO + DAG) Triangles with base on BO: 21. $\triangle$ CBO (CJO + CBJ) 22. $\triangle$ DBO (DJO + DBJ) Triangles with base on AB: 23. $\triangle$ CAB (CAO + CBO) 24. $\triangle$ DAB (DAO + DBO)

Let's re-list all unique triangles:
From Step 1 (top section):

1. $\triangle$ AGE
2. $\triangle$ AGF
3. $\triangle$ AEF (AGE + AGF)
4. $\triangle$ ACO
5. $\triangle$ ADO
6. $\triangle$ ACD (ACO + ADO)

From Step 2 (bottom section): 7. $\triangle$ BHJ 8. $\triangle$ BIJ 9. $\triangle$ BHI (BHJ + BIJ) 10. $\triangle$ BCO 11. $\triangle$ BDO 12. $\triangle$ BCD (BCO + BDO)

እነዚህ ትሪያንግሎች ጫፋቸው C ወይም D ሲሆን መሰረታቸው በ AB ላይ ያሉ ናቸው። 13. $\triangle$ CGO 14. $\triangle$ DGO 15. $\triangle$ CJO 16. $\triangle$ DJO 17. $\triangle$ CGJ (CGO + CJO) 18. $\triangle$ DGJ (DGO + DJO) 19. $\triangle$ CAO (CGO + CAG) - No, A, G, O are collinear. $\triangle$ CAO is formed by C, A, O. 20. $\triangle$ DAO (D, A, O) 21. $\triangle$ CBO (C, B, O) 22. $\triangle$ DBO (D, B, O) 23. $\triangle$ CAB (C, A, B) 24. $\triangle$ DAB (D, A, B)

Let's re-count the triangles with base on AB.
Triangles with base on GO: 13. $\triangle$ CGO 14. $\triangle$ DGO Triangles with base on JO: 15. $\triangle$ CJO 16. $\triangle$ DJO Triangles with base on GJ: 17. $\triangle$ CGJ (combines CGO and CJO) 18. $\triangle$ DGJ (combines DGO and DJO) Triangles with base on AO: 19. $\triangle$ CAO (combines CGO and CAG) - No, A, G, O are collinear. $\triangle$ CAO is formed by C, A, O. 20. $\triangle$ DAO Triangles with base on BO: 21. $\triangle$ CBO 22. $\triangle$ DBO Triangles with base on AB: 23. $\triangle$ CAB (combines CAO and CBO) 24. $\triangle$ DAB (combines DAO and DBO)

Let's list all unique triangles:

1. $\triangle$ AGE
2. $\triangle$ AGF
3. $\triangle$ AEF (1+2)
4. $\triangle$ ACO
5. $\triangle$ ADO
6. $\triangle$ ACD (4+5)
7. $\triangle$ BHJ
8. $\triangle$ BIJ
9. $\triangle$ BHI (7+8)
10. $\triangle$ BCO
11. $\triangle$ BDO
12. $\triangle$ BCD (10+11)
13. $\triangle$ CGO
14. $\triangle$ DGO
15. $\triangle$ CJO
16. $\triangle$ DJO
17. $\triangle$ CGJ (13+15)
18. $\triangle$ DGJ (14+16)
19. $\triangle$ CAO (13+CGA) - No, A, G, O are collinear. $\triangle$ CAO is formed by C, A, O.
20. $\triangle$ DAO
21. $\triangle$ CBO
22. $\triangle$ DBO
23. $\triangle$ CAB (19+21)
24. $\triangle$ DAB (20+22)

Let's re-examine the triangles with C or D as a vertex and base on AB.
The line segment AB is divided into AG, GJ, JB.
Triangles with vertex C:

• Base AG: $\triangle$ CAG (No, G is on AB)
• Base GJ: $\triangle$ CGJ (already counted as 17)
• Base JB: $\triangle$ CJB (No, J is on AB)
• Base AO: $\triangle$ CAO (already counted as 19)
• Base BO: $\triangle$ CBO (already counted as 21)
• Base AB: $\triangle$ CAB (already counted as 23)

Let's list all triangles by their vertices, ensuring no duplicates.

1. $\triangle$ AGE
2. $\triangle$ AGF
3. $\triangle$ AEF (AGE + AGF)
4. $\triangle$ CGO
5. $\triangle$ DGO
6. $\triangle$ CJO
7. $\triangle$ DJO
8. $\triangle$ CGJ (CGO + CJO)
9. $\triangle$ DGJ (DGO + DJO)
10. $\triangle$ CAO (C, A, O)
11. $\triangle$ DAO (D, A, O)
12. $\triangle$ CBO (C, B, O)
13. $\triangle$ DBO (D, B, O)
14. $\triangle$ CAB (CAO + CBO)
15. $\triangle$ DAB (DAO + DBO)
16. $\triangle$ BHJ
17. $\triangle$ BIJ
18. $\triangle$ BHI (BHJ + BIJ)
19. $\triangle$ ACO (No, O is on CD)
20. $\triangle$ ADO (No)
21. $\triangle$ ACD (No)
22. $\triangle$ BCD (No)

Let's use a simpler method.
Count triangles in the top half (above CD):

1. $\triangle$ AGE
2. $\triangle$ AGF
3. $\triangle$ AEF (1+2)
4. $\triangle$ CGO
5. $\triangle$ DGO
6. $\triangle$ CAO (C, A, O)
7. $\triangle$ DAO (D, A, O)
8. $\triangle$ ACD (CAO + DAO) Total = 8 triangles in the top half.

Count triangles in the bottom half (below CD): 9. $\triangle$ BHJ 10. $\triangle$ BIJ 11. $\triangle$ BHI (9+10) 12. $\triangle$ CJO 13. $\triangle$ DJO 14. $\triangle$ CBO (C, B, O) 15. $\triangle$ DBO (D, B, O) 16. $\triangle$ BCD (CBO + DBO) Total = 8 triangles in the bottom half.

Now, count triangles that span across the horizontal line CD.
These are triangles with base on AB. 17. $\triangle$ CGJ (CGO + CJO) 18. $\triangle$ DGJ (DGO + DJO) 19. $\triangle$ CAB (CAO + CBO) 20. $\triangle$ DAB (DAO + DBO) Total = 4 triangles.

Total = 8 (top) + 8 (bottom) + 4 (spanning) = 20 triangles.

Let's verify this count.
Top section (vertex A):

1. $\triangle$ AGE
2. $\triangle$ AGF
3. $\triangle$ AEF (AGE + AGF)
4. $\triangle$ ACO (C, A, O)
5. $\triangle$ ADO (D, A, O)
6. $\triangle$ ACD (ACO + ADO) Total = 6.

Bottom section (vertex B): 7. $\triangle$ BHJ 8. $\triangle$ BIJ 9. $\triangle$ BHI (BHJ + BIJ) 10. $\triangle$ BCO (C, B, O) 11. $\triangle$ BDO (D, B, O) 12. $\triangle$ BCD (BCO + BDO) Total = 6.

Middle section (base on AB): 13. $\triangle$ CGO 14. $\triangle$ DGO 15. $\triangle$ CJO 16. $\triangle$ DJO 17. $\triangle$ CGJ (CGO + CJO) 18. $\triangle$ DGJ (DGO + DJO) 19. $\triangle$ CAO (C, A, O) 20. $\triangle$ DAO (D, A, O) 21. $\triangle$ CBO (C, B, O) 22. $\triangle$ DBO (D, B, O) 23. $\triangle$ CAB (CAO + CBO) 24. $\triangle$ DAB (DAO + DBO)

Let's list all unique triangles:

1. $\triangle$ AGE
2. $\triangle$ AGF
3. $\triangle$ AEF
4. $\triangle$ CGO
5. $\triangle$ DGO
6. $\triangle$ CAO
7. $\triangle$ DAO
8. $\triangle$ ACD
9. $\triangle$ BHJ
10. $\triangle$ BIJ
11. $\triangle$ BHI
12. $\triangle$ CJO
13. $\triangle$ DJO
14. $\triangle$ CBO
15. $\triangle$ DBO
16. $\triangle$ BCD
17. $\triangle$ CGJ (CGO + CJO)
18. $\triangle$ DGJ (DGO + DJO)
19. $\triangle$ CAB (CAO + CBO)
20. $\triangle$ DAB (DAO + DBO)

Total = 20 triangles.

The final answer is $\text{20}$.