prove basic proportional theorem
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given triangle ABC where DE||BC
To prove AD/DE =AE/EC
cons. join BE and CD draw DM perpendicular AC and EN perpendicular AB
proof ar( ADE )half ×base×height
half ×AD×EN
ar (BDE ) half ×base×height
half ×DB×EN
divide 1and2
ar ADE /ar BDE =half×AD×EN /half×DB×EN
ar ADE /ar BDE =AD/DB
ar ADE /ar DEC =AE/ECN
∆BDE and∆DEC are on the same base DE between same parallel BC and DE
hence AD/DB =AE /EC
To prove AD/DE =AE/EC
cons. join BE and CD draw DM perpendicular AC and EN perpendicular AB
proof ar( ADE )half ×base×height
half ×AD×EN
ar (BDE ) half ×base×height
half ×DB×EN
divide 1and2
ar ADE /ar BDE =half×AD×EN /half×DB×EN
ar ADE /ar BDE =AD/DB
ar ADE /ar DEC =AE/ECN
∆BDE and∆DEC are on the same base DE between same parallel BC and DE
hence AD/DB =AE /EC
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