Applying the scheme

Let us see the scheme at work on a complete problem, following the six steps one by one.

Example — Applying the scheme

Problem: In the triangle $ABC$, let $M$ be the midpoint of $BC$. Prove that $AM < \dfrac{AB+AC}{2}$.

1. Figure: triangle $ABC$ with $M$ the midpoint of $BC$. I extend $AM$ by a segment $MD\cong AM$.

2. Hyp: $BM\cong MC$.    Thes: $AM < \dfrac{AB+AC}{2}$.

3. Backward Thinking: it is enough to show $2\,AM < AB+AC$, that is $AM+MD < AB+AC$, that is $AD < AB+AC$.

4. Construction: I extend $AM$ beyond $M$ by $MD = AM$ and I join $D$ to $C$.

5. Triangles: $\triangle AMB$ and $\triangle DMC$ have $AM\cong MD$, $BM\cong MC$, $\widehat{AMB}\cong\widehat{DMC}$ (vertically opposite) $\implies$ by the first criterion $\implies$ $AB\cong DC$.

6. Conclusion: By the triangle inequality in $\triangle ADC$: $AD < AC + DC = AC + AB$. Therefore $2\,AM = AD < AB + AC$, that is $AM < \dfrac{AB+AC}{2}$. $\blacksquare$

Links

Topics: Euclidean geometry
Concepts: Congruence criteria · Proof · Triangle inequality · Median
Skills: Proving · Synthetic geometry