Question

Please help to solve this question, and provide the formal proof. Very much appreciated, and have a great day!

Answer 1

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Answer:

NB = b

NC = c

Explanation:

A "quick and dirty" proof could be this:

1. M is the midpoint of AN; M is the midpoint of BC . . . . given
2. ABNC is a parallelogram . . . . diagonals AN and BC have the same midpoint (M)*
3. NB = AC = b; NC = AB = c . . . . opposite sides of a parallelogram are congruent.

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More formally, we might prove it like this:

1. MC = MB . . . . definition of median
2. MA = MN, AC = b, AB = c . . . . given
3. ∠CMA ≅ ∠BMN . . . . vertical angles are congruent
4. ΔCMA ≅ ΔBMN . . . . SAS congruence postulate
5. AC ≅ NB . . . . CPCTC
6. NB = b . . . . substitution property of equality/congruence
7. ∠CMN ≅ ∠ BMA . . . . vertical angles are congruent
8. ΔCMN ≅ ΔBMA . . . . SAS congruence postulate
9. AB ≅ NC . . . . CPCTC
10. NC = c . . . . substitution property of equality

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* The proof is "dirty" only in that the theorem "if the midpoints of the diagonals of a quadrilateral are coincident, then that quadrilateral is a parallelogram" is not one usually available to cite in proofs. The "more formal" proof here essentially proves that the quadrilateral is a parallelogram.