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Question Number 79266 by Maclaurin Stickker last updated on 24/Jan/20

let ABC be a escalene triangle of  area 7. Let A_1  be a point on the side  BC, and let B_1  and C_1  be points on  the sides AC and AB, such that  AA_1 , BB_1  and CC_1  are parallel. Find  the area of triangle A_1 B_1 C_1 .

$${let}\:{ABC}\:{be}\:{a}\:{escalene}\:{triangle}\:{of} \\ $$$${area}\:\mathrm{7}.\:{Let}\:{A}_{\mathrm{1}} \:{be}\:{a}\:{point}\:{on}\:{the}\:{side} \\ $$$${BC},\:{and}\:{let}\:{B}_{\mathrm{1}} \:{and}\:{C}_{\mathrm{1}} \:{be}\:{points}\:{on} \\ $$$${the}\:{sides}\:{AC}\:{and}\:{AB},\:{such}\:{that} \\ $$$${AA}_{\mathrm{1}} ,\:{BB}_{\mathrm{1}} \:{and}\:{CC}_{\mathrm{1}} \:{are}\:{parallel}.\:{Find} \\ $$$${the}\:{area}\:{of}\:{triangle}\:{A}_{\mathrm{1}} {B}_{\mathrm{1}} {C}_{\mathrm{1}} . \\ $$

Commented by mr W last updated on 24/Jan/20

Δ_(A_1 B_1 C_1 ) =2Δ_(ABC)   2×7=14

$$\Delta_{{A}_{\mathrm{1}} {B}_{\mathrm{1}} {C}_{\mathrm{1}} } =\mathrm{2}\Delta_{{ABC}} \\ $$$$\mathrm{2}×\mathrm{7}=\mathrm{14} \\ $$

Commented by mr W last updated on 24/Jan/20

Commented by mr W last updated on 24/Jan/20

Δ_(AB_1 C_1 ) =Δ_(ABC)   Δ_(AA_1 B_1 ) =Δ_(AA_1 B)   Δ_(AA_1 C_1 ) =Δ_(AA_1 C)   ⇒Δ_(A_1 B_1 C_1 ) =2Δ_(ABC)

$$\Delta_{{AB}_{\mathrm{1}} {C}_{\mathrm{1}} } =\Delta_{{ABC}} \\ $$$$\Delta_{{AA}_{\mathrm{1}} {B}_{\mathrm{1}} } =\Delta_{{AA}_{\mathrm{1}} {B}} \\ $$$$\Delta_{{AA}_{\mathrm{1}} {C}_{\mathrm{1}} } =\Delta_{{AA}_{\mathrm{1}} {C}} \\ $$$$\Rightarrow\Delta_{{A}_{\mathrm{1}} {B}_{\mathrm{1}} {C}_{\mathrm{1}} } =\mathrm{2}\Delta_{{ABC}} \\ $$

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