the foot of altitudes in a triangle is given, how to find the vertices?

Oh Boy! Great question!! I liked it!!!

Pick a paper and pencil and sketch the following diagram. Original triangle is ABC. Le AD be perp to BC, BE perp to AC and CF perp to AB, They will be naturally concurrent say at H, the ortho-center. Join D, E and F.

Now B, F, Eand C are concyclic so angle FEB = Angle FCB = 90-B = angle BED because A, E, D and B are concyclic. So Angle FED = 180 - 2B.Now by finding DE, EF and FD you can use cosine rule and find angle FED and so angle B. Now in similar way you can calculate angles A and C.Angle FCD or FCB = 90-B and angle EBD = 90 - C so construct circles on FD of angle 90-B and on ED construct a circle of angle 90-C these 2 circles are loci of C and B respectively.

These circles intersect at D of course but other point of intersection is A. Now Join AD and draw the perp to AD at D. Where it meets the circle are B and C.