In triangle abc ad and bc are altitudes prove that ar( dec)/ar( adc)= dc2/ac2
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see diagram.
AD and BE are altitudes. ∠AEA = ∠BDA and ∠DAC = ∠DBC.
Hence, quadrilateral ABDE is a cyclic quad. Hence opposite angles are supplementary,
∠AED = 180° - ∠B
=> ∠DEC = ∠B
=> in ΔDEC, ∠EDC = ∠A
Hence, the two triangles ABC and DEC are similar. Thus the areas of the triangles are proportional to the squares of corresponding sides.
=> Ar (DEC) / Ar (ABC) = DC² / AC² or DE² / AB²
Also note that the quadrilateral CDHE is also cyclic.
AD and BE are altitudes. ∠AEA = ∠BDA and ∠DAC = ∠DBC.
Hence, quadrilateral ABDE is a cyclic quad. Hence opposite angles are supplementary,
∠AED = 180° - ∠B
=> ∠DEC = ∠B
=> in ΔDEC, ∠EDC = ∠A
Hence, the two triangles ABC and DEC are similar. Thus the areas of the triangles are proportional to the squares of corresponding sides.
=> Ar (DEC) / Ar (ABC) = DC² / AC² or DE² / AB²
Also note that the quadrilateral CDHE is also cyclic.
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