1. **Problem Statement:** Count the number of triangles in the given geometric figure. 2. **Understanding the figure:** The figure is a large triangle with vertices at bottom-left (A), top (B), and bottom-right (C). Inside, there are 7 points total, including the vertices and points on the segments: - A vertical line from B to midpoint D of base AC. - A horizontal line from A to a point E on BD. - Two diagonal lines: from B to a point F on AC, and from F to C. 3. **Approach:** We count all triangles formed by these points and segments. 4. **Step-by-step counting:** - Triangles formed by the large triangle ABC itself: 1 - Triangles formed by the vertical line BD splitting ABC into ABD and BDC: 2 - Triangles formed by horizontal line AE inside ABD: 2 more (ABE and AED) - Triangles formed by diagonal lines BF and FC inside BDC: 2 more (BFC and FGC) - Smaller triangles formed by intersections of these lines inside the figure: - Triangle AEF - Triangle EFD - Triangle FDC Counting all distinct triangles: - ABC - ABD - BDC - ABE - AED - BFC - FGC - AEF - EFD - FDC Total triangles = 10 5. **Final answer:** $$\boxed{10}$$