How many triangles are there in the following figure?

Concept: When several straight lines meet at one point (an apex) and land on n distinct points of a straight base line, every pair of those base points, taken together with the apex, is automatically a triangle, because the two connecting lines and the base segment between them are already drawn. That gives C(n, 2) triangles from the fan alone. On top of the fan, separately look for (a) smaller triangles carved out by any extra diagonal cutting across a fan-triangle, and (b) larger triangles that appear only because two segments are actually one continuous straight line through a shared vertex, letting a side run further than it first looks. A reliable count always checks all three layers, and never stops at the smallest visible pieces.

Applying it here:

1. Five straight lines run from the apex down to five points on the base line (the two outer corners and three points in between). Any 2 of these 5 points, taken with the apex, form a triangle whose sides are already drawn, giving C(5, 2) = 10 triangles from this fan alone.

2. Above each of those 4 base segments, one diagonal rises from its left end and meets the cevian standing over its right end at a point partway up that cevian, not at the base itself. That touch-point is a new, distinct vertex, so the diagonal splits the one-segment fan-triangle above each base segment into two different smaller triangles: a lower one (diagonal + the short stretch of the cevian below the touch-point + the base segment) and an upper one (diagonal + the apex + the remaining stretch of the same cevian above the touch-point, ending at the touch-point rather than at the far base point). Because each of these 8 triangles uses a touch-point instead of a base point, none of them repeats any triangle already counted in Step 1. Four segments times 2 gives 8 further triangles.

3. Below the base line, the middle base point drops a straight line to a point beneath the figure, and the two outer base corners also run straight down to that same point, forming an inverted triangle that is itself split into two halves by that middle vertical line: 3 triangles in this region (the two halves plus the whole).

4. The line from the apex to the middle base point does not stop at the base; it continues, unbroken, all the way down to the point below. Because it is genuinely one straight line, each outer edge of the upper figure combines with it and with the matching lower edge to trace one full-height triangle running from the apex to the bottom point: 2 more triangles that are easy to miss if the upper and lower halves are counted separately.

5. Total: 10 + 8 + 3 + 2 = 23.

Cross-check: Grouping the same figure by size instead of location gives an identical, independently-checked total with no overlap between groups: the 8 zig-zag triangles (each using one of the 4 diagonal touch-points as a vertex), the 10 apex-fan triangles (built only from the 5 base points, none of which include a touch-point), the 3 triangles below the base, and the 2 full-height triangles. That is 8 + 10 + 3 + 2 = 23 again, with every triangle counted in one group only, confirming the total independently.

So the figure contains 23 triangles in total.