The coordinates of the feet of the perpendiculars from the vertices of a triangle on the opposite sides are $(20, 25), (8, 16) and (8, 9)$. Find the sum of y-co-ordinates of all vertices of the triangle?

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.

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

$FD=20\text{ and }EF=7$

So that $h=\frac{7\times 20+20\times 8+15\times 8}{7+20+15}=10$

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

Thus the coordinates of 0 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

$y-9=\frac{-10-8}{15-16}(x-8)$

$\Rightarrow 3y+x-35=0...\left(ii\right)$

and equation of BC is $y-25=-\frac{20-10}{25-15}(x-20)$

$\Rightarrow y+x-45=0...\left(iii\right)$

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