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). The coordinates of a vertex of the triangle are
(5, 10)
(50, -5)
(15, 30)
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=√(20−8)2+(25−16)2=15
FD=20 and EF=7
so that h=7×20+20×8+15×87+20+15=10
and k=7×25+20×16+15×97+20+15=15
thus, the coordinates of O are (10,15)
Since AC is perpendicular to OE, equation of AC is
y−16=−10−815−6(x−8)⇒ y−2x=0 (1)
Similarly equation of AB is
y−9=−10−815−9(x−8)⇒ 3y+x−35=0 (2)
and equation of BC is
y−25=−20−1025−15(x−20)⇒ y+x−45=0 (3)
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)