In △ABC, D, E, and F are the midpoints of sides AB, BC, and CA respectively. Show that △DEF divides △ABC into four congruent triangles.

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GIVEN: D, E, F are midpoints of AB, BC, CA of △ABC. TO PROVE: △DEF divides △ABC into four congruent triangles. PROOF using Mid-Point Theorem: STEP 1 — Apply Mid-Point Theorem three times: (a) In △ABC, D is midpoint of AB and F is midpoint of CA. By Mid-Point Theorem: DF ∥ BC and DF = BC/2. (b) In △ABC, D is midpoint of AB and E is midpoint of BC. By Mid-Point Theorem: DE ∥ CA and DE = CA/2. (c) In △ABC, E is midpoint of BC and F is midpoint of CA. By Mid-Point Theorem: EF ∥ AB and EF = AB/2. STEP 2 — Four triangles formed: The four triangles are △ADF, △DBE, △FEC, and △DEF. STEP 3 — Show congruence: Consider △ADF and △FED: AD = DF (AD = AB/2 = DF by Step 1(a)). Wait — let me use direct approach. AD = FE (AD = AB/2, FE = AB/2 from Step 1(c) — EF = AB/2). DF = BE (DF = BC/2, BE = BC/2). AF = DE (AF = CA/2, DE = CA/2). So by SSS: △ADF ≅ △FED ≅ △DBE ≅ △EFC. All four triangles are congruent (SSS). CONCLUSION: △DEF divides △ABC into 4 congruent triangles, each with sides equal to half the original triangle's sides. Each small triangle has 1/4 the area of △ABC.

Practice — Quadrilaterals

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