Show that by joining the midpoints of an equilateral triangle, we divide it into 4 identical equilateral triangles. [Hint: Note that the corner triangles are isosceles.]
Answer
Let the triangle be ABC with every side of length a, and let P, Q, R be the midpoints of BC, CA, AB.
Joining the three midpoints makes the shaded middle triangle PQR and three corner triangles.
The three corner triangles. Take triangle ARQ. Since R and Q are midpoints,
AR = AB⁄2 = a⁄2 and AQ = AC⁄2 = a⁄2 So triangle ARQ is isosceles with AR = AQ, and the angle between them is ∠A = 60°. Base angles = (180° − 60°)⁄2 = 60° each.
All three angles are 60°, so ARQ is equilateral with side a⁄2. The same argument at B and at C makes BRP and CQP equilateral with side a⁄2.
The middle triangle. RQ joins the midpoints of AB and AC, so by the midpoint theorem RQ = BC⁄2 = a⁄2, and likewise QP = a⁄2 and PR = a⁄2. So PQR is equilateral with side a⁄2 too.
All four triangles are equilateral with side a⁄2, hence identical (congruent).
Why this matters for the fractal: because the four pieces are identical and each is an exact half-scale copy of the original, removing the middle one and repeating the construction gives a shape that looks the same at every scale. Self-similarity is built in from this one fact.