Sides $AB$ and $AC$ and median $AD$ to $ riangle ABC$ are respectively proportional to sides $PQ$ and $PR$ and median $PM$ of another triangle $PQR$. Show that $ riangle ABC \sim riangle PQR$.

Question

Sides $AB$ and $AC$ and median $AD$ to $	riangle ABC$ are respectively proportional to sides $PQ$ and $PR$ and median $PM$ of another triangle $PQR$. Show that $	riangle ABC \sim 	riangle PQR$.

Accepted Answer

Step 1: Two triangles ABC and PQR with medians AD and PM, extended to E and N respectively
Step 2: Produce $AD$ to $E$ such that $AD = DE$ and join $EC$. Produce $PM$ to $N$ such that $PM = MN$ and join $NR$.
Step 3: $	riangle ADB \cong 	riangle EDC$ (SAS) $\Rightarrow AB = EC$. Similarly $PQ = NR$.
Step 4: Since $\frac{AB}{PQ} = \frac{AC}{PR} = \frac{AD}{PM}$, we get $\frac{EC}{NR} = \frac{AC}{PR} = \frac{AE}{PN}$
$	herefore 	riangle AEC \sim 	riangle PNR \Rightarrow \angle 1 = \angle 2$
Step 5: Similarly $\angle 3 = \angle 4$. Hence $\angle 1 + \angle 3 = \angle 2 + \angle 4$, i.e., $\angle A = \angle P$.
Also $\frac{AB}{PQ} = \frac{AC}{PR}$. $	herefore 	riangle ABC \sim 	riangle PQR$ (SAS similarity)

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