Trisection points divide the base into three equal segments, and both triangles share the same vertex altitude. Area equals (1/2) x base x height, so their areas are identical. Straight dissection along altitudes enables rearrangement.
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By Pythagoras theorem, leg semicircles sum to the hypotenuse semicircle: Semicircle(a) + Semicircle(b) = Semicircle(c). Subtracting the two unshaded circular segments from both sides shows that the lunes’ area Area(A) + Area(B) equals the triangle’s area Area(C).
The fraction of the rectangle covered by n identical tangent circles in a row is always constant at π / 4 approximately 11/14 (or 78.5%) regardless of whether n equals 10, 20 or 50.
Drawing a diagonal splits the trapezium into two triangles having bases a and b, each sharing perpendicular height h. Summing their areas gives (1/2)ah + (1/2)bh = (1/2)(a + b)h.
The trapezium is split into a parallelogram of base a and a triangle of base (b – a), both sharing height h. Area = ah + (1/2)(b – a)h = (1/2)(2a + b – a)h = (1/2)(a + b)h.