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The altitude form the vertices A, B and...

The altitude form the vertices A, B and C of the triangle ABC meet its circumcircle at D,E and F, respectively . The complex number representing the points D,E, and F are `z_(1),z_(2)` and `z_(3)`, respectively. If `(z_(3) -z_(1))//(z_(2) -z_(1))` is purely real, then show that triangle ABC is right-angled at A.

Text Solution

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`angle FDA = angle FCA = 90^(@) - A`
`angleADE = angle ABF = 90^(@) - A`
`rArr angleFDE = 180^(@) -2A = 2pi - 2A`
Simpliarly `angleDFE= 2pi - 2C and angle DEF = 2pi - 2B`
The angle of `DeltaDEF` are `pi -2A, pi -2B` and `pi -2C`, respectively
Also it is given that `(z_(3) - z_(1)) //(z_(2)-z_(1))` is purely real. Hence,
`arg((z_(3) -z_(1))/(z_(2)-z_(1))) = 0 or pi`
`rArr pi - 2A = 0 or pi`
`rArr A= (pi)/(2) or 0` (not permissible)
Hence, triangle ABC is right angled at A.
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