Question Number 79266 by Maclaurin Stickker last updated on 24/Jan/20
$${let}\:{ABC}\:{be}\:{a}\:{escalene}\:{triangle}\:{of} \\ $$$${area}\:\mathrm{7}.\:{Let}\:{A}_{\mathrm{1}} \:{be}\:{a}\:{point}\:{on}\:{the}\:{side} \\ $$$${BC},\:{and}\:{let}\:{B}_{\mathrm{1}} \:{and}\:{C}_{\mathrm{1}} \:{be}\:{points}\:{on} \\ $$$${the}\:{sides}\:{AC}\:{and}\:{AB},\:{such}\:{that} \\ $$$${AA}_{\mathrm{1}} ,\:{BB}_{\mathrm{1}} \:{and}\:{CC}_{\mathrm{1}} \:{are}\:{parallel}.\:{Find} \\ $$$${the}\:{area}\:{of}\:{triangle}\:{A}_{\mathrm{1}} {B}_{\mathrm{1}} {C}_{\mathrm{1}} . \\ $$
Commented by mr W last updated on 24/Jan/20
$$\Delta_{{A}_{\mathrm{1}} {B}_{\mathrm{1}} {C}_{\mathrm{1}} } =\mathrm{2}\Delta_{{ABC}} \\ $$$$\mathrm{2}×\mathrm{7}=\mathrm{14} \\ $$
Commented by mr W last updated on 24/Jan/20
Commented by mr W last updated on 24/Jan/20
$$\Delta_{{AB}_{\mathrm{1}} {C}_{\mathrm{1}} } =\Delta_{{ABC}} \\ $$$$\Delta_{{AA}_{\mathrm{1}} {B}_{\mathrm{1}} } =\Delta_{{AA}_{\mathrm{1}} {B}} \\ $$$$\Delta_{{AA}_{\mathrm{1}} {C}_{\mathrm{1}} } =\Delta_{{AA}_{\mathrm{1}} {C}} \\ $$$$\Rightarrow\Delta_{{A}_{\mathrm{1}} {B}_{\mathrm{1}} {C}_{\mathrm{1}} } =\mathrm{2}\Delta_{{ABC}} \\ $$