ExampleConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + claude-sonnet-5)audited 2026-08-17

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The four midpoint subtriangles of the $0, 1, i$ triangle display all three cancelling interior edges

Example

For $a = 0$ , $b = 1$ , and $c = i$ , the side midpoints are

$p = \frac{1}{2}, q = \frac{1 + i}{2}, r = \frac{i}{2} .$

The midpoint subdivision, with the orientation of Filled complex triangles, their oriented three-edge boundary contours, diameter, and perimeter, consists of $Δ [0, p, r]$ , $Δ [p, 1, q]$ , $Δ [r, q, i]$ , and $Δ [p, q, r]$ .

Facts & Assumptions

Given: The displayed vertices and midpoints, with every triangle carrying the orientation fixed by its ordered vertices.

[L1]

If $p, q, r$ are the side midpoints of $Δ [a, b, c]$ and $f$ is continuous on that filled triangle, then $I_{f} [a, b, c] = I_{f} [a, p, r] + I_{f} [p, b, q] + I_{f} [r, q, c] + I_{f} [p, q, r]$ (Midpoint subdivision of a triangle cancels every interior edge and preserves its outer boundary integral).

Verification

technique · direct

1.1

The directed edge lists are $(0 \to p, p \to r, r \to 0)$ , $(p \to 1, 1 \to q, q \to p)$ , $(r \to q, q \to i, i \to r)$ , and $(p \to q, q \to r, r \to p)$ .

given

2.1

The interior segment pairs are $p \to r, r \to p$ , $q \to p, p \to q$ , and $r \to q, q \to r$ ; each pair consists of one directed edge and its reversal, so the formal oriented edges cancel.

step 1.1algebra

3.1

The surviving half-edges concatenate as $0 \to p \to 1$ , $1 \to q \to i$ , and $i \to r \to 0$ , which is the positive outer boundary $0 \to 1 \to i \to 0$ ; for every continuous integrand, this is exactly the integral identity in [L1].

step 1.1step 2.1L1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 39 results over 11 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

E. Stein and R. Shakarchi, Complex Analysis, Ch. 2, Theorem 1.1 (standard reference, not scraped)

The four midpoint subtriangles of the 0,1,i0,1,i triangle display all three cancelling interior edges