1. **Stating the problem:** We have butterfly-shaped diagrams with values at vertices labeled as $A, B, C, D, E, F, G, H$ arranged in two adjoining triangles. We want to find relationships and solve for these values. 2. **Understanding the butterfly pattern:** The butterfly consists of two triangles sharing a center. The vertices are arranged as: - Top vertices: $A$ and $C$ - Bottom vertices: $F$ and $H$ - Center vertices: $B$ and $G$ - Additional vertices: $D$ and $E$ 3. **Given values for the first example:** $A=3$, $B=20$, $C=5$, $D=26$, $E=14$, $F=4$, $G=6$, $H=2$ 4. **Formula and rules:** The butterfly pattern often follows the rule: $$D = A \times B$$ $$E = C \times G$$ $$B + G = F + H$$ 5. **Check the first formula:** Calculate $A \times B = 3 \times 20 = 60$ but given $D=26$, so this is not direct multiplication. 6. **Check the sum of center vertices equals sum of bottom vertices:** $$B + G = 20 + 6 = 26$$ $$F + H = 4 + 2 = 6$$ They are not equal, so this rule does not hold here. 7. **Try another approach: sum of top vertices times sum of bottom vertices equals sum of center vertices times sum of additional vertices:** Calculate: $$ (A + C) \times (F + H) = (3 + 5) \times (4 + 2) = 8 \times 6 = 48$$ $$ (B + G) \times (D + E) = (20 + 6) \times (26 + 14) = 26 \times 40 = 1040$$ Not equal. 8. **Try difference of products:** Calculate: $$D - A \times B = 26 - 3 \times 20 = 26 - 60 = -34$$ $$E - C \times G = 14 - 5 \times 6 = 14 - 30 = -16$$ No clear pattern. 9. **Try sum of products of opposite vertices:** Calculate: $$A \times H + C \times F = 3 \times 2 + 5 \times 4 = 6 + 20 = 26$$ Given $D=26$, matches. Calculate: $$B \times H + G \times F = 20 \times 2 + 6 \times 4 = 40 + 24 = 64$$ Given $E=14$, no match. 10. **Try sum of products of adjacent vertices:** Calculate: $$A \times F + C \times H = 3 \times 4 + 5 \times 2 = 12 + 10 = 22$$ Given $D=26$, no match. 11. **Try sum of products of center vertices equals sum of products of top and bottom vertices:** Calculate: $$B \times G = 20 \times 6 = 120$$ $$A \times C + F \times H = 3 \times 5 + 4 \times 2 = 15 + 8 = 23$$ No match. 12. **Conclusion:** The butterfly pattern here is that $D$ equals the sum of products of opposite vertices $A \times H + C \times F$. 13. **Final formula:** $$D = A \times H + C \times F$$ 14. **Verification:** $$26 = 3 \times 2 + 5 \times 4 = 6 + 20 = 26$$ Correct. 15. **Similarly for $E$:** Try: $$E = B \times H + G \times F$$ Calculate: $$20 \times 2 + 6 \times 4 = 40 + 24 = 64$$ Given $E=14$, no match. Try: $$E = B + G$$ $$20 + 6 = 26$$ No match. Try: $$E = B - G$$ $$20 - 6 = 14$$ Matches given $E=14$. 16. **Summary of relations:** $$D = A \times H + C \times F$$ $$E = B - G$$ 17. **Explanation:** - $D$ is the sum of products of opposite vertices in the butterfly. - $E$ is the difference between the two center vertices. 18. **Apply to other examples similarly.** **Final answer:** $$D = A \times H + C \times F$$ $$E = B - G$$