Consider the following figure : [Diagram shows a composite geometric f

We can count the triangles by listing unique combinations of 3 non-collinear vertices from the set {A, B, C, D, E, F, G}.
Collinear sets of 3 points: (A, F, G), (D, F, E), (B, G, C).

Let’s list the triangles systematically:
1. Triangles with A as a vertex:
* Bases on DE: ADF, AFE, ADE (3)
* Bases on BC: ABG, ACG, ABC (3)
* Bases connecting D/E to G: ADG, AEG (2)
* Bases connecting B/C to F: ABF, ACF (2)
(Total from A = 3+3+2+2 = 10 unique triangles with A as apex).

2. Triangles with F as a vertex (excluding those with A as apex, already counted):
* Bases on BC: FBG, FCG, FBC (3)
* Bases connecting B/C to D/E: FBD, FBE, FCD, FCE (4)
(Total from F, excluding A as apex = 3+4 = 7 unique triangles with F as apex, or not having A as apex).

3. Triangles with D as a vertex (excluding those with A or F as apex):
* Bases on BC: DBG, DCG, DBC (3)
* Bases connecting G/C to E: DGE, DCE (2)
(Total from D, excluding A or F as apex = 3+2 = 5 unique triangles).

4. Triangles with E as a vertex (excluding those with A, F, or D as apex):
* Bases on BC: EBG, ECG, EBC (3)
(Total from E, excluding A, F, D as apex = 3 unique triangles).

5. Triangles with B as a vertex (excluding those with A, F, D, E as apex):
* Bases connecting D/E to G/E/C: BDE (1)
(Total from B, new = 1 unique triangle).

6. Triangles with C as a vertex (excluding those with A, F, D, E, B as apex):
* Bases connecting D/E to G/D/B: CDE (1)
(Total from C, new = 1 unique triangle).

Total unique triangles = (Count from A) + (New from F) + (New from D) + (New from E) + (New from B) + (New from C)
Total = 10 + 7 + 5 + 3 + 1 + 1 = 27.

Let’s list the 27 unique triangles based on this count:
From A (10): ADF, AFE, ADE, ABF, ACF, ABG, ACG, ABC, ADG, AEG
From F (7): FBG, FCG, FBC, FBD, FBE, FCD, FCE
From D (5): DBG, DCG, DBC, DGE, DCE
From E (3): EBG, ECG, EBC
From B (1): BDE
From C (1): CDE

Checking for duplicates between the groups:
– FBD is same as BDF (BDF in B list)
– FCE is same as CEF (CEF in C list)
– FBE is new.
– FCD is new.
– DBG is same as GDB.
– DCG is same as GDC.
– DBC is same as CDB.
– DGE is same as EDG.
– DCE is same as EDC.
– EBG is same as GEB.
– ECG is same as GEC.
– EBC is same as BCE.
– BDE is new.
– CDE is new.

Let’s refine the list by listing unique triangle names:
ADF, AFE, ADE, ABF, ACF, ABG, ACG, ABC, ADG, AEG (10)
FBG, FCG, FBC, FBE, FCD (5 new from F, excluding BDF, CEF which are listed separately)
DBG, DCG, DBC, DGE, DCE (5 new from D)
EBG, ECG, EBC (3 new from E)
BDF (1 new from B)
BDE (1 new from B)
CEF (1 new from C)
CDE (1 new from C)

Total = 10 + 5 + 5 + 3 + 1 + 1 + 1 + 1 = 29? No, this is not 27. Re-checking the grouping.

Let’s use the approach based on counting regions.
10 smallest regions: ADF, AFE, BDF, CEF, FBG, FCG, DBG, DCG, EBG, ECG (10 triangles).
Combinations of 2 smallest regions: ADE, ABF, ACF, FBC, DBC, EBC, ADG, AEG, DGE, DCE, BDE, CDE (12 triangles).
Combinations of 3 smallest regions: None obvious.
Combinations of 4 smallest regions: ABG, ACG (2 triangles).
Combinations of 8 smallest regions: ABC (1 triangle).
Total from regions: 10 + 12 + 2 + 1 = 25.

There must be triangles formed by vertices that are not simple combinations of adjacent smallest regions in this manner. The vertex-based counting method seems more reliable.

Let’s re-verify the vertex count of 27.
A (10): ADF, AFE, ADE, ABF, ACF, ABG, ACG, ABC, ADG, AEG
F (7): FBG, FCG, FBC, FBD, FBE, FCD, FCE
D (6): DBG, DCG, DBC, DGE, DCE, DFB
E (6): EBG, ECG, EBC, EDG, EDC, EFC
B (3): BDE, BEF, BDF
C (3): CDE, CFD, CFE
G (8): GDB, GDC, GEB, GEC, GFB, GFC, GAD, GAE (same as DBG, DCG, EBG, ECG, FBG, FCG, ADG, AEG)

Unique list from combined lists:
ADF, AFE, ADE, ABF, ACF, ABG, ACG, ABC, ADG, AEG (10)
FBG, FCG, FBC (3)
FBD (BDF) (1)
FBE (1)
FCD (1)
FCE (CEF) (1)
DBG, DCG, DBC (3)
DGE (1)
DCE (1)
EBG, ECG, EBC (3)
BDE (1)
CDE (1)

Total: 10 + 3 + 1 + 1 + 1 + 1 + 3 + 1 + 1 + 3 + 1 + 1 = 29.

Let’s re-check the list of 29 again for strict uniqueness.
1. ADF
2. AFE
3. ADE (ADF+AFE)
4. ABF (ADF+BDF)
5. ACF (AFE+CEF)
6. ABG (ADF+BDF+FBG+DBG – regions) or (ABF+FBG?) No. (A,B,G)
7. ACG (A,C,G)
8. ABC (A,B,C)
9. ADG (A,D,G)
10. AEG (A,E,G)
11. FBG
12. FCG
13. FBC (FBG+FCG)
14. BDF
15. CEF
16. DBG
17. DCG
18. DBC (DBG+DCG)
19. EBG
20. ECG
21. EBC (EBG+ECG)
22. DGE
23. DCE
24. BDE
25. CDE
26. FBE
27. FCD

Summing these unique names: 27.

Let’s re-verify the count of 27 based on groups which summed to 27.
A (10): ADF, AFE, ADE, ABF, ACF, ABG, ACG, ABC, ADG, AEG
F (7): FBG, FCG, FBC, FBD, FBE, FCD, FCE. (No A)
D (5): DBG, DCG, DBC, DGE, DCE. (No A, F as apex)
E (3): EBG, ECG, EBC. (No A, F, D as apex)
B (1): BDE. (No A, F, D, E as apex)
C (1): CDE. (No A, F, D, E, B as apex)

Let’s check for overlaps between these groups:
A and F: No triangle from F list has A as apex. OK.
A and D: No triangle from D list has A as apex. OK.
A and E: No triangle from E list has A as apex. OK.
A and B: BDE has B as highest vertex. OK.
A and C: CDE has C as highest vertex. OK.

F and D: FBD(F,B,D) vs DBG(D,B,G), DCG(D,C,G), DBC(D,B,C), DGE(D,G,E), DCE(D,C,E). BDF from B-list. FBD from F-list. Yes, FBD=BDF. So, when listing from F, FBD is F,B,D. From D, it’s D,B,F. From B, it’s B,D,F.
My list of 27 seems correct. Let’s proceed with 27.