The coordinates of the feet of the perpendicular from the vertices of triangle on the opposite sides are $(20, 25),(8, 16) and (8, 9)$. The coordinates of a vertex of the triangle are 

We use the fact that the orthocentre O of the triangle ABC is the incentre of the pedal triangle DEF. Let $(h,k)$ be the coordinates of O.

$\text{ Now }ED=\sqrt{(20-8{)}^2+(25-16{)}^2}=15$

$FD = 20 and EF = 7$

$\text{ so that }h=\frac{7\times 20+20\times 8+15\times 8}{7+20+15}=10\text{ and }$

$k=\frac{7\times 25+20\times 16+15\times 9}{7+20+15}=15$

Thus the coordinates of O are $(10,15)$Since AC is perpendicular to OE, equation of AC is

$y-16=-\frac{10-8}{15-16}(x-8)\Rightarrow y-2x=0.....\left(i\right)$

Similarly equation of AB is

$y-9=\frac{10-8}{15-9}(x-8)\Rightarrow 3y+x-35=0.....\left(ii\right)$

and equation of BC is 

$y-25=\frac{20-10}{25-15}(x-20)\Rightarrow y+x-45=0....\left(iii\right)$

Solving (1) and (2) we get$A(5,10).$
From (2) and (3) we get $B(50,–5)$and from (3) and (1) we get $C(15,30)$